Article MCART_3, MML version 4.99.1005

:: MCART_3:th 1
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9, b10 being set
             st b3 in b4 & b4 in b5 & b5 in b6 & b6 in b7 & b7 in b8 & b8 in b9 & b9 in b10 & b10 in b2
          holds b3 misses b1);

:: MCART_3:th 2
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set
             st b3 in b4 & b4 in b5 & b5 in b6 & b6 in b7 & b7 in b8 & b8 in b9 & b9 in b10 & b10 in b11 & b11 in b2
          holds b3 misses b1);

:: MCART_3:funcnot 1 => MCART_3:func 1
definition
  let a1, a2, a3, a4, a5, a6 be set;
  func [A1,A2,A3,A4,A5,A6] -> set equals
    [[a1,a2,a3,a4,a5],a6];
end;

:: MCART_3:def 1
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[b1,b2,b3,b4,b5,b6] = [[b1,b2,b3,b4,b5],b6];

:: MCART_3:th 3
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[b1,b2,b3,b4,b5,b6] = [[[[[b1,b2],b3],b4],b5],b6];

:: MCART_3:th 4
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[b1,b2,b3,b4,b5,b6] = [[b1,b2,b3,b4],b5,b6];

:: MCART_3:th 5
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[b1,b2,b3,b4,b5,b6] = [[b1,b2,b3],b4,b5,b6];

:: MCART_3:th 6
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[b1,b2,b3,b4,b5,b6] = [[b1,b2],b3,b4,b5,b6];

:: MCART_3:th 7
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12 being set
      st [b1,b2,b3,b4,b5,b6] = [b7,b8,b9,b10,b11,b12]
   holds b1 = b7 & b2 = b8 & b3 = b9 & b4 = b10 & b5 = b11 & b6 = b12;

:: MCART_3:th 8
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8 being set
             st (b3 in b1 or b4 in b1)
          holds b2 <> [b3,b4,b5,b6,b7,b8]);

:: MCART_3:funcnot 2 => MCART_3:func 2
definition
  let a1, a2, a3, a4, a5, a6 be set;
  func [:A1,A2,A3,A4,A5,A6:] -> set equals
    [:[:a1,a2,a3,a4,a5:],a6:];
end;

:: MCART_3:def 2
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[:b1,b2,b3,b4,b5,b6:] = [:[:b1,b2,b3,b4,b5:],b6:];

:: MCART_3:th 9
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[:b1,b2,b3,b4,b5,b6:] = [:[:[:[:[:b1,b2:],b3:],b4:],b5:],b6:];

:: MCART_3:th 10
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[:b1,b2,b3,b4,b5,b6:] = [:[:b1,b2,b3,b4:],b5,b6:];

:: MCART_3:th 11
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[:b1,b2,b3,b4,b5,b6:] = [:[:b1,b2,b3:],b4,b5,b6:];

:: MCART_3:th 12
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[:b1,b2,b3,b4,b5,b6:] = [:[:b1,b2:],b3,b4,b5,b6:];

:: MCART_3:th 13
theorem
for b1, b2, b3, b4, b5, b6 being set holds
   b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
iff
   [:b1,b2,b3,b4,b5,b6:] <> {};

:: MCART_3:th 14
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12 being set
      st b1 <> {} &
         b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         [:b1,b2,b3,b4,b5,b6:] = [:b7,b8,b9,b10,b11,b12:]
   holds b1 = b7 & b2 = b8 & b3 = b9 & b4 = b10 & b5 = b11 & b6 = b12;

:: MCART_3:th 15
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12 being set
      st [:b1,b2,b3,b4,b5,b6:] <> {} &
         [:b1,b2,b3,b4,b5,b6:] = [:b7,b8,b9,b10,b11,b12:]
   holds b1 = b7 & b2 = b8 & b3 = b9 & b4 = b10 & b5 = b11 & b6 = b12;

:: MCART_3:th 16
theorem
for b1, b2 being set
      st [:b1,b1,b1,b1,b1,b1:] = [:b2,b2,b2,b2,b2,b2:]
   holds b1 = b2;

:: MCART_3:th 17
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:] holds
   ex b8 being Element of b1 st
      ex b9 being Element of b2 st
         ex b10 being Element of b3 st
            ex b11 being Element of b4 st
               ex b12 being Element of b5 st
                  ex b13 being Element of b6 st
                     b7 = [b8,b9,b10,b11,b12,b13];

:: MCART_3:funcnot 3 => MCART_3:func 3
definition
  let a1, a2, a3, a4, a5, a6 be set;
  let a7 be Element of [:a1,a2,a3,a4,a5,a6:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {};
  func A7 `1 -> Element of a1 means
    for b1, b2, b3, b4, b5, b6 being set
          st a7 = [b1,b2,b3,b4,b5,b6]
       holds it = b1;
end;

:: MCART_3:def 3
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:]
for b8 being Element of b1 holds
      b8 = b7 `1
   iff
      for b9, b10, b11, b12, b13, b14 being set
            st b7 = [b9,b10,b11,b12,b13,b14]
         holds b8 = b9;

:: MCART_3:funcnot 4 => MCART_3:func 4
definition
  let a1, a2, a3, a4, a5, a6 be set;
  let a7 be Element of [:a1,a2,a3,a4,a5,a6:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {};
  func A7 `2 -> Element of a2 means
    for b1, b2, b3, b4, b5, b6 being set
          st a7 = [b1,b2,b3,b4,b5,b6]
       holds it = b2;
end;

:: MCART_3:def 4
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:]
for b8 being Element of b2 holds
      b8 = b7 `2
   iff
      for b9, b10, b11, b12, b13, b14 being set
            st b7 = [b9,b10,b11,b12,b13,b14]
         holds b8 = b10;

:: MCART_3:funcnot 5 => MCART_3:func 5
definition
  let a1, a2, a3, a4, a5, a6 be set;
  let a7 be Element of [:a1,a2,a3,a4,a5,a6:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {};
  func A7 `3 -> Element of a3 means
    for b1, b2, b3, b4, b5, b6 being set
          st a7 = [b1,b2,b3,b4,b5,b6]
       holds it = b3;
end;

:: MCART_3:def 5
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:]
for b8 being Element of b3 holds
      b8 = b7 `3
   iff
      for b9, b10, b11, b12, b13, b14 being set
            st b7 = [b9,b10,b11,b12,b13,b14]
         holds b8 = b11;

:: MCART_3:funcnot 6 => MCART_3:func 6
definition
  let a1, a2, a3, a4, a5, a6 be set;
  let a7 be Element of [:a1,a2,a3,a4,a5,a6:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {};
  func A7 `4 -> Element of a4 means
    for b1, b2, b3, b4, b5, b6 being set
          st a7 = [b1,b2,b3,b4,b5,b6]
       holds it = b4;
end;

:: MCART_3:def 6
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:]
for b8 being Element of b4 holds
      b8 = b7 `4
   iff
      for b9, b10, b11, b12, b13, b14 being set
            st b7 = [b9,b10,b11,b12,b13,b14]
         holds b8 = b12;

:: MCART_3:funcnot 7 => MCART_3:func 7
definition
  let a1, a2, a3, a4, a5, a6 be set;
  let a7 be Element of [:a1,a2,a3,a4,a5,a6:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {};
  func A7 `5 -> Element of a5 means
    for b1, b2, b3, b4, b5, b6 being set
          st a7 = [b1,b2,b3,b4,b5,b6]
       holds it = b5;
end;

:: MCART_3:def 7
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:]
for b8 being Element of b5 holds
      b8 = b7 `5
   iff
      for b9, b10, b11, b12, b13, b14 being set
            st b7 = [b9,b10,b11,b12,b13,b14]
         holds b8 = b13;

:: MCART_3:funcnot 8 => MCART_3:func 8
definition
  let a1, a2, a3, a4, a5, a6 be set;
  let a7 be Element of [:a1,a2,a3,a4,a5,a6:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {};
  func A7 `6 -> Element of a6 means
    for b1, b2, b3, b4, b5, b6 being set
          st a7 = [b1,b2,b3,b4,b5,b6]
       holds it = b6;
end;

:: MCART_3:def 8
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:]
for b8 being Element of b6 holds
      b8 = b7 `6
   iff
      for b9, b10, b11, b12, b13, b14 being set
            st b7 = [b9,b10,b11,b12,b13,b14]
         holds b8 = b14;

:: MCART_3:th 18
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:]
for b8, b9, b10, b11, b12, b13 being set
      st b7 = [b8,b9,b10,b11,b12,b13]
   holds b7 `1 = b8 & b7 `2 = b9 & b7 `3 = b10 & b7 `4 = b11 & b7 `5 = b12 & b7 `6 = b13;

:: MCART_3:th 19
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:] holds
   b7 = [b7 `1,b7 `2,b7 `3,b7 `4,b7 `5,b7 `6];

:: MCART_3:th 20
theorem
for b1, b2, b3, b4, b5, b6 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b7 being Element of [:b1,b2,b3,b4,b5,b6:] holds
   b7 `1 = b7 `1 `1 `1 `1 `1 &
    b7 `2 = b7 `1 `1 `1 `1 `2 &
    b7 `3 = b7 `1 `1 `1 `2 &
    b7 `4 = b7 `1 `1 `2 &
    b7 `5 = b7 `1 `2 &
    b7 `6 = b7 `2;

:: MCART_3:th 21
theorem
for b1, b2, b3, b4, b5, b6 being set
      st (not b1 c= [:b1,b2,b3,b4,b5,b6:] & not b1 c= [:b2,b3,b4,b5,b6,b1:] & not b1 c= [:b3,b4,b5,b6,b1,b2:] & not b1 c= [:b4,b5,b6,b1,b2,b3:] & not b1 c= [:b5,b6,b1,b2,b3,b4:] implies b1 c= [:b6,b1,b2,b3,b4,b5:])
   holds b1 = {};

:: MCART_3:th 22
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12 being set
      st [:b1,b2,b3,b4,b5,b6:] meets [:b7,b8,b9,b10,b11,b12:]
   holds b1 meets b7 & b2 meets b8 & b3 meets b9 & b4 meets b10 & b5 meets b11 & b6 meets b12;

:: MCART_3:th 23
theorem
for b1, b2, b3, b4, b5, b6 being set holds
[:{b1},{b2},{b3},{b4},{b5},{b6}:] = {[b1,b2,b3,b4,b5,b6]};

:: MCART_3:th 24
theorem
for b1, b2, b3, b4, b5, b6 being set
for b7 being Element of [:b1,b2,b3,b4,b5,b6:]
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {}
for b8, b9, b10, b11, b12, b13 being set
      st b7 = [b8,b9,b10,b11,b12,b13]
   holds b7 `1 = b8 & b7 `2 = b9 & b7 `3 = b10 & b7 `4 = b11 & b7 `5 = b12 & b7 `6 = b13;

:: MCART_3:th 25
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Element of [:b2,b3,b4,b5,b6,b7:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         (for b9 being Element of b2
         for b10 being Element of b3
         for b11 being Element of b4
         for b12 being Element of b5
         for b13 being Element of b6
         for b14 being Element of b7
               st b8 = [b9,b10,b11,b12,b13,b14]
            holds b1 = b9)
   holds b1 = b8 `1;

:: MCART_3:th 26
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Element of [:b2,b3,b4,b5,b6,b7:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         (for b9 being Element of b2
         for b10 being Element of b3
         for b11 being Element of b4
         for b12 being Element of b5
         for b13 being Element of b6
         for b14 being Element of b7
               st b8 = [b9,b10,b11,b12,b13,b14]
            holds b1 = b10)
   holds b1 = b8 `2;

:: MCART_3:th 27
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Element of [:b2,b3,b4,b5,b6,b7:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         (for b9 being Element of b2
         for b10 being Element of b3
         for b11 being Element of b4
         for b12 being Element of b5
         for b13 being Element of b6
         for b14 being Element of b7
               st b8 = [b9,b10,b11,b12,b13,b14]
            holds b1 = b11)
   holds b1 = b8 `3;

:: MCART_3:th 28
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Element of [:b2,b3,b4,b5,b6,b7:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         (for b9 being Element of b2
         for b10 being Element of b3
         for b11 being Element of b4
         for b12 being Element of b5
         for b13 being Element of b6
         for b14 being Element of b7
               st b8 = [b9,b10,b11,b12,b13,b14]
            holds b1 = b12)
   holds b1 = b8 `4;

:: MCART_3:th 29
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Element of [:b2,b3,b4,b5,b6,b7:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         (for b9 being Element of b2
         for b10 being Element of b3
         for b11 being Element of b4
         for b12 being Element of b5
         for b13 being Element of b6
         for b14 being Element of b7
               st b8 = [b9,b10,b11,b12,b13,b14]
            holds b1 = b13)
   holds b1 = b8 `5;

:: MCART_3:th 30
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Element of [:b2,b3,b4,b5,b6,b7:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         (for b9 being Element of b2
         for b10 being Element of b3
         for b11 being Element of b4
         for b12 being Element of b5
         for b13 being Element of b6
         for b14 being Element of b7
               st b8 = [b9,b10,b11,b12,b13,b14]
            holds b1 = b14)
   holds b1 = b8 `6;

:: MCART_3:th 31
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
      st b1 in [:b2,b3,b4,b5,b6,b7:]
   holds ex b8, b9, b10, b11, b12, b13 being set st
      b8 in b2 & b9 in b3 & b10 in b4 & b11 in b5 & b12 in b6 & b13 in b7 & b1 = [b8,b9,b10,b11,b12,b13];

:: MCART_3:th 32
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12 being set holds
   [b1,b2,b3,b4,b5,b6] in [:b7,b8,b9,b10,b11,b12:]
iff
   b1 in b7 & b2 in b8 & b3 in b9 & b4 in b10 & b5 in b11 & b6 in b12;

:: MCART_3:th 33
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
      st for b8 being set holds
              b8 in b7
           iff
              ex b9, b10, b11, b12, b13, b14 being set st
                 b9 in b1 & b10 in b2 & b11 in b3 & b12 in b4 & b13 in b5 & b14 in b6 & b8 = [b9,b10,b11,b12,b13,b14]
   holds b7 = [:b1,b2,b3,b4,b5,b6:];

:: MCART_3:th 34
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {} & b10 <> {} & b11 <> {} & b12 <> {}
for b13 being Element of [:b1,b2,b3,b4,b5,b6:]
for b14 being Element of [:b7,b8,b9,b10,b11,b12:]
      st b13 = b14
   holds b13 `1 = b14 `1 & b13 `2 = b14 `2 & b13 `3 = b14 `3 & b13 `4 = b14 `4 & b13 `5 = b14 `5 & b13 `6 = b14 `6;

:: MCART_3:th 35
theorem
for b1, b2, b3, b4, b5, b6 being set
for b7 being Element of bool b1
for b8 being Element of bool b2
for b9 being Element of bool b3
for b10 being Element of bool b4
for b11 being Element of bool b5
for b12 being Element of bool b6
for b13 being Element of [:b1,b2,b3,b4,b5,b6:]
      st b13 in [:b7,b8,b9,b10,b11,b12:]
   holds b13 `1 in b7 & b13 `2 in b8 & b13 `3 in b9 & b13 `4 in b10 & b13 `5 in b11 & b13 `6 in b12;

:: MCART_3:th 36
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12 being set
      st b1 c= b7 & b2 c= b8 & b3 c= b9 & b4 c= b10 & b5 c= b11 & b6 c= b12
   holds [:b1,b2,b3,b4,b5,b6:] c= [:b7,b8,b9,b10,b11,b12:];

:: MCART_3:th 37
theorem
for b1, b2, b3, b4, b5, b6 being set
for b7 being Element of bool b1
for b8 being Element of bool b2
for b9 being Element of bool b3
for b10 being Element of bool b4
for b11 being Element of bool b5
for b12 being Element of bool b6 holds
   [:b7,b8,b9,b10,b11,b12:] is Element of bool [:b1,b2,b3,b4,b5,b6:];

:: MCART_3:th 38
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9, b10, b11, b12 being set
             st b3 in b4 & b4 in b5 & b5 in b6 & b6 in b7 & b7 in b8 & b8 in b9 & b9 in b10 & b10 in b11 & b11 in b12 & b12 in b2
          holds b3 misses b1);

:: MCART_3:th 39
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13 being set
             st b3 in b4 & b4 in b5 & b5 in b6 & b6 in b7 & b7 in b8 & b8 in b9 & b9 in b10 & b10 in b11 & b11 in b12 & b12 in b13 & b13 in b2
          holds b3 misses b1);

:: MCART_3:funcnot 9 => MCART_3:func 9
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  func [A1,A2,A3,A4,A5,A6,A7] -> set equals
    [[a1,a2,a3,a4,a5,a6],a7];
end;

:: MCART_3:def 9
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2,b3,b4,b5,b6],b7];

:: MCART_3:th 40
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[b1,b2,b3,b4,b5,b6,b7] = [[[[[[b1,b2],b3],b4],b5],b6],b7];

:: MCART_3:th 41
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2,b3,b4,b5],b6,b7];

:: MCART_3:th 42
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2,b3,b4],b5,b6,b7];

:: MCART_3:th 43
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2,b3],b4,b5,b6,b7];

:: MCART_3:th 44
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2],b3,b4,b5,b6,b7];

:: MCART_3:th 45
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14 being set
      st [b1,b2,b3,b4,b5,b6,b7] = [b8,b9,b10,b11,b12,b13,b14]
   holds b1 = b8 & b2 = b9 & b3 = b10 & b4 = b11 & b5 = b12 & b6 = b13 & b7 = b14;

:: MCART_3:th 46
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9 being set
             st (b3 in b1 or b4 in b1)
          holds b2 <> [b3,b4,b5,b6,b7,b8,b9]);

:: MCART_3:funcnot 10 => MCART_3:func 10
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  func [:A1,A2,A3,A4,A5,A6,A7:] -> set equals
    [:[:a1,a2,a3,a4,a5,a6:],a7:];
end;

:: MCART_3:def 10
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2,b3,b4,b5,b6:],b7:];

:: MCART_3:th 47
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:[:[:[:[:b1,b2:],b3:],b4:],b5:],b6:],b7:];

:: MCART_3:th 48
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2,b3,b4,b5:],b6,b7:];

:: MCART_3:th 49
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2,b3,b4:],b5,b6,b7:];

:: MCART_3:th 50
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2,b3:],b4,b5,b6,b7:];

:: MCART_3:th 51
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2:],b3,b4,b5,b6,b7:];

:: MCART_3:th 52
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
   b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
iff
   [:b1,b2,b3,b4,b5,b6,b7:] <> {};

:: MCART_3:th 53
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14 being set
      st b1 <> {} &
         b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         [:b1,b2,b3,b4,b5,b6,b7:] = [:b8,b9,b10,b11,b12,b13,b14:]
   holds b1 = b8 & b2 = b9 & b3 = b10 & b4 = b11 & b5 = b12 & b6 = b13 & b7 = b14;

:: MCART_3:th 54
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14 being set
      st [:b1,b2,b3,b4,b5,b6,b7:] <> {} &
         [:b1,b2,b3,b4,b5,b6,b7:] = [:b8,b9,b10,b11,b12,b13,b14:]
   holds b1 = b8 & b2 = b9 & b3 = b10 & b4 = b11 & b5 = b12 & b6 = b13 & b7 = b14;

:: MCART_3:th 55
theorem
for b1, b2 being set
      st [:b1,b1,b1,b1,b1,b1,b1:] = [:b2,b2,b2,b2,b2,b2,b2:]
   holds b1 = b2;

:: MCART_3:th 56
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:] holds
   ex b9 being Element of b1 st
      ex b10 being Element of b2 st
         ex b11 being Element of b3 st
            ex b12 being Element of b4 st
               ex b13 being Element of b5 st
                  ex b14 being Element of b6 st
                     ex b15 being Element of b7 st
                        b8 = [b9,b10,b11,b12,b13,b14,b15];

:: MCART_3:funcnot 11 => MCART_3:func 11
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  let a8 be Element of [:a1,a2,a3,a4,a5,a6,a7:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {};
  func A8 `1 -> Element of a1 means
    for b1, b2, b3, b4, b5, b6, b7 being set
          st a8 = [b1,b2,b3,b4,b5,b6,b7]
       holds it = b1;
end;

:: MCART_3:def 11
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
for b9 being Element of b1 holds
      b9 = b8 `1
   iff
      for b10, b11, b12, b13, b14, b15, b16 being set
            st b8 = [b10,b11,b12,b13,b14,b15,b16]
         holds b9 = b10;

:: MCART_3:funcnot 12 => MCART_3:func 12
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  let a8 be Element of [:a1,a2,a3,a4,a5,a6,a7:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {};
  func A8 `2 -> Element of a2 means
    for b1, b2, b3, b4, b5, b6, b7 being set
          st a8 = [b1,b2,b3,b4,b5,b6,b7]
       holds it = b2;
end;

:: MCART_3:def 12
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
for b9 being Element of b2 holds
      b9 = b8 `2
   iff
      for b10, b11, b12, b13, b14, b15, b16 being set
            st b8 = [b10,b11,b12,b13,b14,b15,b16]
         holds b9 = b11;

:: MCART_3:funcnot 13 => MCART_3:func 13
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  let a8 be Element of [:a1,a2,a3,a4,a5,a6,a7:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {};
  func A8 `3 -> Element of a3 means
    for b1, b2, b3, b4, b5, b6, b7 being set
          st a8 = [b1,b2,b3,b4,b5,b6,b7]
       holds it = b3;
end;

:: MCART_3:def 13
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
for b9 being Element of b3 holds
      b9 = b8 `3
   iff
      for b10, b11, b12, b13, b14, b15, b16 being set
            st b8 = [b10,b11,b12,b13,b14,b15,b16]
         holds b9 = b12;

:: MCART_3:funcnot 14 => MCART_3:func 14
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  let a8 be Element of [:a1,a2,a3,a4,a5,a6,a7:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {};
  func A8 `4 -> Element of a4 means
    for b1, b2, b3, b4, b5, b6, b7 being set
          st a8 = [b1,b2,b3,b4,b5,b6,b7]
       holds it = b4;
end;

:: MCART_3:def 14
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
for b9 being Element of b4 holds
      b9 = b8 `4
   iff
      for b10, b11, b12, b13, b14, b15, b16 being set
            st b8 = [b10,b11,b12,b13,b14,b15,b16]
         holds b9 = b13;

:: MCART_3:funcnot 15 => MCART_3:func 15
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  let a8 be Element of [:a1,a2,a3,a4,a5,a6,a7:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {};
  func A8 `5 -> Element of a5 means
    for b1, b2, b3, b4, b5, b6, b7 being set
          st a8 = [b1,b2,b3,b4,b5,b6,b7]
       holds it = b5;
end;

:: MCART_3:def 15
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
for b9 being Element of b5 holds
      b9 = b8 `5
   iff
      for b10, b11, b12, b13, b14, b15, b16 being set
            st b8 = [b10,b11,b12,b13,b14,b15,b16]
         holds b9 = b14;

:: MCART_3:funcnot 16 => MCART_3:func 16
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  let a8 be Element of [:a1,a2,a3,a4,a5,a6,a7:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {};
  func A8 `6 -> Element of a6 means
    for b1, b2, b3, b4, b5, b6, b7 being set
          st a8 = [b1,b2,b3,b4,b5,b6,b7]
       holds it = b6;
end;

:: MCART_3:def 16
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
for b9 being Element of b6 holds
      b9 = b8 `6
   iff
      for b10, b11, b12, b13, b14, b15, b16 being set
            st b8 = [b10,b11,b12,b13,b14,b15,b16]
         holds b9 = b15;

:: MCART_3:funcnot 17 => MCART_3:func 17
definition
  let a1, a2, a3, a4, a5, a6, a7 be set;
  let a8 be Element of [:a1,a2,a3,a4,a5,a6,a7:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {};
  func A8 `7 -> Element of a7 means
    for b1, b2, b3, b4, b5, b6, b7 being set
          st a8 = [b1,b2,b3,b4,b5,b6,b7]
       holds it = b7;
end;

:: MCART_3:def 17
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
for b9 being Element of b7 holds
      b9 = b8 `7
   iff
      for b10, b11, b12, b13, b14, b15, b16 being set
            st b8 = [b10,b11,b12,b13,b14,b15,b16]
         holds b9 = b16;

:: MCART_3:th 57
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
for b9, b10, b11, b12, b13, b14, b15 being set
      st b8 = [b9,b10,b11,b12,b13,b14,b15]
   holds b8 `1 = b9 & b8 `2 = b10 & b8 `3 = b11 & b8 `4 = b12 & b8 `5 = b13 & b8 `6 = b14 & b8 `7 = b15;

:: MCART_3:th 58
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:] holds
   b8 = [b8 `1,b8 `2,b8 `3,b8 `4,b8 `5,b8 `6,b8 `7];

:: MCART_3:th 59
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:] holds
   b8 `1 = b8 `1 `1 `1 `1 `1 `1 &
    b8 `2 = b8 `1 `1 `1 `1 `1 `2 &
    b8 `3 = b8 `1 `1 `1 `1 `2 &
    b8 `4 = b8 `1 `1 `1 `2 &
    b8 `5 = b8 `1 `1 `2 &
    b8 `6 = b8 `1 `2 &
    b8 `7 = b8 `2;

:: MCART_3:th 60
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
      st (not b1 c= [:b1,b2,b3,b4,b5,b6,b7:] & not b1 c= [:b2,b3,b4,b5,b6,b7,b1:] & not b1 c= [:b3,b4,b5,b6,b7,b1,b2:] & not b1 c= [:b4,b5,b6,b7,b1,b2,b3:] & not b1 c= [:b5,b6,b7,b1,b2,b3,b4:] & not b1 c= [:b6,b7,b1,b2,b3,b4,b5:] implies b1 c= [:b7,b1,b2,b3,b4,b5,b6:])
   holds b1 = {};

:: MCART_3:th 61
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14 being set
      st [:b1,b2,b3,b4,b5,b6,b7:] meets [:b8,b9,b10,b11,b12,b13,b14:]
   holds b1 meets b8 & b2 meets b9 & b3 meets b10 & b4 meets b11 & b5 meets b12 & b6 meets b13 & b7 meets b14;

:: MCART_3:th 62
theorem
for b1, b2, b3, b4, b5, b6, b7 being set holds
[:{b1},{b2},{b3},{b4},{b5},{b6},{b7}:] = {[b1,b2,b3,b4,b5,b6,b7]};

:: MCART_3:th 63
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {}
for b9, b10, b11, b12, b13, b14, b15 being set
      st b8 = [b9,b10,b11,b12,b13,b14,b15]
   holds b8 `1 = b9 & b8 `2 = b10 & b8 `3 = b11 & b8 `4 = b12 & b8 `5 = b13 & b8 `6 = b14 & b8 `7 = b15;

:: MCART_3:th 64
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of [:b2,b3,b4,b5,b6,b7,b8:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         (for b10 being Element of b2
         for b11 being Element of b3
         for b12 being Element of b4
         for b13 being Element of b5
         for b14 being Element of b6
         for b15 being Element of b7
         for b16 being Element of b8
               st b9 = [b10,b11,b12,b13,b14,b15,b16]
            holds b1 = b10)
   holds b1 = b9 `1;

:: MCART_3:th 65
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of [:b2,b3,b4,b5,b6,b7,b8:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         (for b10 being Element of b2
         for b11 being Element of b3
         for b12 being Element of b4
         for b13 being Element of b5
         for b14 being Element of b6
         for b15 being Element of b7
         for b16 being Element of b8
               st b9 = [b10,b11,b12,b13,b14,b15,b16]
            holds b1 = b11)
   holds b1 = b9 `2;

:: MCART_3:th 66
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of [:b2,b3,b4,b5,b6,b7,b8:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         (for b10 being Element of b2
         for b11 being Element of b3
         for b12 being Element of b4
         for b13 being Element of b5
         for b14 being Element of b6
         for b15 being Element of b7
         for b16 being Element of b8
               st b9 = [b10,b11,b12,b13,b14,b15,b16]
            holds b1 = b12)
   holds b1 = b9 `3;

:: MCART_3:th 67
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of [:b2,b3,b4,b5,b6,b7,b8:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         (for b10 being Element of b2
         for b11 being Element of b3
         for b12 being Element of b4
         for b13 being Element of b5
         for b14 being Element of b6
         for b15 being Element of b7
         for b16 being Element of b8
               st b9 = [b10,b11,b12,b13,b14,b15,b16]
            holds b1 = b13)
   holds b1 = b9 `4;

:: MCART_3:th 68
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of [:b2,b3,b4,b5,b6,b7,b8:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         (for b10 being Element of b2
         for b11 being Element of b3
         for b12 being Element of b4
         for b13 being Element of b5
         for b14 being Element of b6
         for b15 being Element of b7
         for b16 being Element of b8
               st b9 = [b10,b11,b12,b13,b14,b15,b16]
            holds b1 = b14)
   holds b1 = b9 `5;

:: MCART_3:th 69
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of [:b2,b3,b4,b5,b6,b7,b8:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         (for b10 being Element of b2
         for b11 being Element of b3
         for b12 being Element of b4
         for b13 being Element of b5
         for b14 being Element of b6
         for b15 being Element of b7
         for b16 being Element of b8
               st b9 = [b10,b11,b12,b13,b14,b15,b16]
            holds b1 = b15)
   holds b1 = b9 `6;

:: MCART_3:th 70
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of [:b2,b3,b4,b5,b6,b7,b8:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         (for b10 being Element of b2
         for b11 being Element of b3
         for b12 being Element of b4
         for b13 being Element of b5
         for b14 being Element of b6
         for b15 being Element of b7
         for b16 being Element of b8
               st b9 = [b10,b11,b12,b13,b14,b15,b16]
            holds b1 = b16)
   holds b1 = b9 `7;

:: MCART_3:th 71
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
      st b1 in [:b2,b3,b4,b5,b6,b7,b8:]
   holds ex b9, b10, b11, b12, b13, b14, b15 being set st
      b9 in b2 & b10 in b3 & b11 in b4 & b12 in b5 & b13 in b6 & b14 in b7 & b15 in b8 & b1 = [b9,b10,b11,b12,b13,b14,b15];

:: MCART_3:th 72
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14 being set holds
   [b1,b2,b3,b4,b5,b6,b7] in [:b8,b9,b10,b11,b12,b13,b14:]
iff
   b1 in b8 & b2 in b9 & b3 in b10 & b4 in b11 & b5 in b12 & b6 in b13 & b7 in b14;

:: MCART_3:th 73
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
      st for b9 being set holds
              b9 in b8
           iff
              ex b10, b11, b12, b13, b14, b15, b16 being set st
                 b10 in b1 & b11 in b2 & b12 in b3 & b13 in b4 & b14 in b5 & b15 in b6 & b16 in b7 & b9 = [b10,b11,b12,b13,b14,b15,b16]
   holds b8 = [:b1,b2,b3,b4,b5,b6,b7:];

:: MCART_3:th 74
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {} & b10 <> {} & b11 <> {} & b12 <> {} & b13 <> {} & b14 <> {}
for b15 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
for b16 being Element of [:b8,b9,b10,b11,b12,b13,b14:]
      st b15 = b16
   holds b15 `1 = b16 `1 & b15 `2 = b16 `2 & b15 `3 = b16 `3 & b15 `4 = b16 `4 & b15 `5 = b16 `5 & b15 `6 = b16 `6 & b15 `7 = b16 `7;

:: MCART_3:th 75
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Element of bool b1
for b9 being Element of bool b2
for b10 being Element of bool b3
for b11 being Element of bool b4
for b12 being Element of bool b5
for b13 being Element of bool b6
for b14 being Element of bool b7
for b15 being Element of [:b1,b2,b3,b4,b5,b6,b7:]
      st b15 in [:b8,b9,b10,b11,b12,b13,b14:]
   holds b15 `1 in b8 & b15 `2 in b9 & b15 `3 in b10 & b15 `4 in b11 & b15 `5 in b12 & b15 `6 in b13 & b15 `7 in b14;

:: MCART_3:th 76
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14 being set
      st b1 c= b8 & b2 c= b9 & b3 c= b10 & b4 c= b11 & b5 c= b12 & b6 c= b13 & b7 c= b14
   holds [:b1,b2,b3,b4,b5,b6,b7:] c= [:b8,b9,b10,b11,b12,b13,b14:];

:: MCART_3:th 77
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Element of bool b1
for b9 being Element of bool b2
for b10 being Element of bool b3
for b11 being Element of bool b4
for b12 being Element of bool b5
for b13 being Element of bool b6
for b14 being Element of bool b7 holds
   [:b8,b9,b10,b11,b12,b13,b14:] is Element of bool [:b1,b2,b3,b4,b5,b6,b7:];

:: MCART_3:th 78
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14 being set
             st b3 in b4 & b4 in b5 & b5 in b6 & b6 in b7 & b7 in b8 & b8 in b9 & b9 in b10 & b10 in b11 & b11 in b12 & b12 in b13 & b13 in b14 & b14 in b2
          holds b3 misses b1);

:: MCART_3:th 79
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15 being set
             st b3 in b4 & b4 in b5 & b5 in b6 & b6 in b7 & b7 in b8 & b8 in b9 & b9 in b10 & b10 in b11 & b11 in b12 & b12 in b13 & b13 in b14 & b14 in b15 & b15 in b2
          holds b3 misses b1);

:: MCART_3:funcnot 18 => MCART_3:func 18
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  func [A1,A2,A3,A4,A5,A6,A7,A8] -> set equals
    [[a1,a2,a3,a4,a5,a6,a7],a8];
end;

:: MCART_3:def 18
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8] = [[b1,b2,b3,b4,b5,b6,b7],b8];

:: MCART_3:th 80
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8] = [[[[[[[b1,b2],b3],b4],b5],b6],b7],b8];

:: MCART_3:th 81
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8] = [[b1,b2,b3,b4,b5,b6],b7,b8];

:: MCART_3:th 82
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8] = [[b1,b2,b3,b4,b5],b6,b7,b8];

:: MCART_3:th 83
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8] = [[b1,b2,b3,b4],b5,b6,b7,b8];

:: MCART_3:th 84
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8] = [[b1,b2,b3],b4,b5,b6,b7,b8];

:: MCART_3:th 85
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8] = [[b1,b2],b3,b4,b5,b6,b7,b8];

:: MCART_3:th 86
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16 being set
      st [b1,b2,b3,b4,b5,b6,b7,b8] = [b9,b10,b11,b12,b13,b14,b15,b16]
   holds b1 = b9 & b2 = b10 & b3 = b11 & b4 = b12 & b5 = b13 & b6 = b14 & b7 = b15 & b8 = b16;

:: MCART_3:th 87
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9, b10 being set
             st (b3 in b1 or b4 in b1)
          holds b2 <> [b3,b4,b5,b6,b7,b8,b9,b10]);

:: MCART_3:funcnot 19 => MCART_3:func 19
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  func [:A1,A2,A3,A4,A5,A6,A7,A8:] -> set equals
    [:[:a1,a2,a3,a4,a5,a6,a7:],a8:];
end;

:: MCART_3:def 19
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8:] = [:[:b1,b2,b3,b4,b5,b6,b7:],b8:];

:: MCART_3:th 88
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8:] = [:[:[:[:[:[:[:b1,b2:],b3:],b4:],b5:],b6:],b7:],b8:];

:: MCART_3:th 89
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8:] = [:[:b1,b2,b3,b4,b5,b6:],b7,b8:];

:: MCART_3:th 90
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8:] = [:[:b1,b2,b3,b4,b5:],b6,b7,b8:];

:: MCART_3:th 91
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8:] = [:[:b1,b2,b3,b4:],b5,b6,b7,b8:];

:: MCART_3:th 92
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8:] = [:[:b1,b2,b3:],b4,b5,b6,b7,b8:];

:: MCART_3:th 93
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8:] = [:[:b1,b2:],b3,b4,b5,b6,b7,b8:];

:: MCART_3:th 94
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
   b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
iff
   [:b1,b2,b3,b4,b5,b6,b7,b8:] <> {};

:: MCART_3:th 95
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16 being set
      st b1 <> {} &
         b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         [:b1,b2,b3,b4,b5,b6,b7,b8:] = [:b9,b10,b11,b12,b13,b14,b15,b16:]
   holds b1 = b9 & b2 = b10 & b3 = b11 & b4 = b12 & b5 = b13 & b6 = b14 & b7 = b15 & b8 = b16;

:: MCART_3:th 96
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16 being set
      st [:b1,b2,b3,b4,b5,b6,b7,b8:] <> {} &
         [:b1,b2,b3,b4,b5,b6,b7,b8:] = [:b9,b10,b11,b12,b13,b14,b15,b16:]
   holds b1 = b9 & b2 = b10 & b3 = b11 & b4 = b12 & b5 = b13 & b6 = b14 & b7 = b15 & b8 = b16;

:: MCART_3:th 97
theorem
for b1, b2 being set
      st [:b1,b1,b1,b1,b1,b1,b1,b1:] = [:b2,b2,b2,b2,b2,b2,b2,b2:]
   holds b1 = b2;

:: MCART_3:th 98
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:] holds
   ex b10 being Element of b1 st
      ex b11 being Element of b2 st
         ex b12 being Element of b3 st
            ex b13 being Element of b4 st
               ex b14 being Element of b5 st
                  ex b15 being Element of b6 st
                     ex b16 being Element of b7 st
                        ex b17 being Element of b8 st
                           b9 = [b10,b11,b12,b13,b14,b15,b16,b17];

:: MCART_3:funcnot 20 => MCART_3:func 20
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  let a9 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {};
  func A9 `1 -> Element of a1 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being set
          st a9 = [b1,b2,b3,b4,b5,b6,b7,b8]
       holds it = b1;
end;

:: MCART_3:def 20
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b10 being Element of b1 holds
      b10 = b9 `1
   iff
      for b11, b12, b13, b14, b15, b16, b17, b18 being set
            st b9 = [b11,b12,b13,b14,b15,b16,b17,b18]
         holds b10 = b11;

:: MCART_3:funcnot 21 => MCART_3:func 21
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  let a9 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {};
  func A9 `2 -> Element of a2 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being set
          st a9 = [b1,b2,b3,b4,b5,b6,b7,b8]
       holds it = b2;
end;

:: MCART_3:def 21
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b10 being Element of b2 holds
      b10 = b9 `2
   iff
      for b11, b12, b13, b14, b15, b16, b17, b18 being set
            st b9 = [b11,b12,b13,b14,b15,b16,b17,b18]
         holds b10 = b12;

:: MCART_3:funcnot 22 => MCART_3:func 22
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  let a9 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {};
  func A9 `3 -> Element of a3 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being set
          st a9 = [b1,b2,b3,b4,b5,b6,b7,b8]
       holds it = b3;
end;

:: MCART_3:def 22
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b10 being Element of b3 holds
      b10 = b9 `3
   iff
      for b11, b12, b13, b14, b15, b16, b17, b18 being set
            st b9 = [b11,b12,b13,b14,b15,b16,b17,b18]
         holds b10 = b13;

:: MCART_3:funcnot 23 => MCART_3:func 23
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  let a9 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {};
  func A9 `4 -> Element of a4 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being set
          st a9 = [b1,b2,b3,b4,b5,b6,b7,b8]
       holds it = b4;
end;

:: MCART_3:def 23
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b10 being Element of b4 holds
      b10 = b9 `4
   iff
      for b11, b12, b13, b14, b15, b16, b17, b18 being set
            st b9 = [b11,b12,b13,b14,b15,b16,b17,b18]
         holds b10 = b14;

:: MCART_3:funcnot 24 => MCART_3:func 24
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  let a9 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {};
  func A9 `5 -> Element of a5 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being set
          st a9 = [b1,b2,b3,b4,b5,b6,b7,b8]
       holds it = b5;
end;

:: MCART_3:def 24
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b10 being Element of b5 holds
      b10 = b9 `5
   iff
      for b11, b12, b13, b14, b15, b16, b17, b18 being set
            st b9 = [b11,b12,b13,b14,b15,b16,b17,b18]
         holds b10 = b15;

:: MCART_3:funcnot 25 => MCART_3:func 25
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  let a9 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {};
  func A9 `6 -> Element of a6 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being set
          st a9 = [b1,b2,b3,b4,b5,b6,b7,b8]
       holds it = b6;
end;

:: MCART_3:def 25
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b10 being Element of b6 holds
      b10 = b9 `6
   iff
      for b11, b12, b13, b14, b15, b16, b17, b18 being set
            st b9 = [b11,b12,b13,b14,b15,b16,b17,b18]
         holds b10 = b16;

:: MCART_3:funcnot 26 => MCART_3:func 26
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  let a9 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {};
  func A9 `7 -> Element of a7 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being set
          st a9 = [b1,b2,b3,b4,b5,b6,b7,b8]
       holds it = b7;
end;

:: MCART_3:def 26
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b10 being Element of b7 holds
      b10 = b9 `7
   iff
      for b11, b12, b13, b14, b15, b16, b17, b18 being set
            st b9 = [b11,b12,b13,b14,b15,b16,b17,b18]
         holds b10 = b17;

:: MCART_3:funcnot 27 => MCART_3:func 27
definition
  let a1, a2, a3, a4, a5, a6, a7, a8 be set;
  let a9 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {};
  func A9 `8 -> Element of a8 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being set
          st a9 = [b1,b2,b3,b4,b5,b6,b7,b8]
       holds it = b8;
end;

:: MCART_3:def 27
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b10 being Element of b8 holds
      b10 = b9 `8
   iff
      for b11, b12, b13, b14, b15, b16, b17, b18 being set
            st b9 = [b11,b12,b13,b14,b15,b16,b17,b18]
         holds b10 = b18;

:: MCART_3:th 99
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b10, b11, b12, b13, b14, b15, b16, b17 being set
      st b9 = [b10,b11,b12,b13,b14,b15,b16,b17]
   holds b9 `1 = b10 & b9 `2 = b11 & b9 `3 = b12 & b9 `4 = b13 & b9 `5 = b14 & b9 `6 = b15 & b9 `7 = b16 & b9 `8 = b17;

:: MCART_3:th 100
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:] holds
   b9 = [b9 `1,b9 `2,b9 `3,b9 `4,b9 `5,b9 `6,b9 `7,b9 `8];

:: MCART_3:th 101
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:] holds
   b9 `1 = b9 `1 `1 `1 `1 `1 `1 `1 &
    b9 `2 = b9 `1 `1 `1 `1 `1 `1 `2 &
    b9 `3 = b9 `1 `1 `1 `1 `1 `2 &
    b9 `4 = b9 `1 `1 `1 `1 `2 &
    b9 `5 = b9 `1 `1 `1 `2 &
    b9 `6 = b9 `1 `1 `2 &
    b9 `7 = b9 `1 `2 &
    b9 `8 = b9 `2;

:: MCART_3:th 102
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
      st (not b1 c= [:b1,b2,b3,b4,b5,b6,b7,b8:] & not b1 c= [:b2,b3,b4,b5,b6,b7,b8,b1:] & not b1 c= [:b3,b4,b5,b6,b7,b8,b1,b2:] & not b1 c= [:b4,b5,b6,b7,b8,b1,b2,b3:] & not b1 c= [:b5,b6,b7,b8,b1,b2,b3,b4:] & not b1 c= [:b6,b7,b8,b1,b2,b3,b4,b5:] & not b1 c= [:b7,b8,b1,b2,b3,b4,b5,b6:] implies b1 c= [:b8,b1,b2,b3,b4,b5,b6,b7:])
   holds b1 = {};

:: MCART_3:th 103
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16 being set
      st [:b1,b2,b3,b4,b5,b6,b7,b8:] meets [:b9,b10,b11,b12,b13,b14,b15,b16:]
   holds b1 meets b9 & b2 meets b10 & b3 meets b11 & b4 meets b12 & b5 meets b13 & b6 meets b14 & b7 meets b15 & b8 meets b16;

:: MCART_3:th 104
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
[:{b1},{b2},{b3},{b4},{b5},{b6},{b7},{b8}:] = {[b1,b2,b3,b4,b5,b6,b7,b8]};

:: MCART_3:th 105
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {}
for b10, b11, b12, b13, b14, b15, b16, b17 being set
      st b9 = [b10,b11,b12,b13,b14,b15,b16,b17]
   holds b9 `1 = b10 & b9 `2 = b11 & b9 `3 = b12 & b9 `4 = b13 & b9 `5 = b14 & b9 `6 = b15 & b9 `7 = b16 & b9 `8 = b17;

:: MCART_3:th 106
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         (for b11 being Element of b2
         for b12 being Element of b3
         for b13 being Element of b4
         for b14 being Element of b5
         for b15 being Element of b6
         for b16 being Element of b7
         for b17 being Element of b8
         for b18 being Element of b9
               st b10 = [b11,b12,b13,b14,b15,b16,b17,b18]
            holds b1 = b11)
   holds b1 = b10 `1;

:: MCART_3:th 107
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         (for b11 being Element of b2
         for b12 being Element of b3
         for b13 being Element of b4
         for b14 being Element of b5
         for b15 being Element of b6
         for b16 being Element of b7
         for b17 being Element of b8
         for b18 being Element of b9
               st b10 = [b11,b12,b13,b14,b15,b16,b17,b18]
            holds b1 = b12)
   holds b1 = b10 `2;

:: MCART_3:th 108
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         (for b11 being Element of b2
         for b12 being Element of b3
         for b13 being Element of b4
         for b14 being Element of b5
         for b15 being Element of b6
         for b16 being Element of b7
         for b17 being Element of b8
         for b18 being Element of b9
               st b10 = [b11,b12,b13,b14,b15,b16,b17,b18]
            holds b1 = b13)
   holds b1 = b10 `3;

:: MCART_3:th 109
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         (for b11 being Element of b2
         for b12 being Element of b3
         for b13 being Element of b4
         for b14 being Element of b5
         for b15 being Element of b6
         for b16 being Element of b7
         for b17 being Element of b8
         for b18 being Element of b9
               st b10 = [b11,b12,b13,b14,b15,b16,b17,b18]
            holds b1 = b14)
   holds b1 = b10 `4;

:: MCART_3:th 110
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         (for b11 being Element of b2
         for b12 being Element of b3
         for b13 being Element of b4
         for b14 being Element of b5
         for b15 being Element of b6
         for b16 being Element of b7
         for b17 being Element of b8
         for b18 being Element of b9
               st b10 = [b11,b12,b13,b14,b15,b16,b17,b18]
            holds b1 = b15)
   holds b1 = b10 `5;

:: MCART_3:th 111
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         (for b11 being Element of b2
         for b12 being Element of b3
         for b13 being Element of b4
         for b14 being Element of b5
         for b15 being Element of b6
         for b16 being Element of b7
         for b17 being Element of b8
         for b18 being Element of b9
               st b10 = [b11,b12,b13,b14,b15,b16,b17,b18]
            holds b1 = b16)
   holds b1 = b10 `6;

:: MCART_3:th 112
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         (for b11 being Element of b2
         for b12 being Element of b3
         for b13 being Element of b4
         for b14 being Element of b5
         for b15 being Element of b6
         for b16 being Element of b7
         for b17 being Element of b8
         for b18 being Element of b9
               st b10 = [b11,b12,b13,b14,b15,b16,b17,b18]
            holds b1 = b17)
   holds b1 = b10 `7;

:: MCART_3:th 113
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         (for b11 being Element of b2
         for b12 being Element of b3
         for b13 being Element of b4
         for b14 being Element of b5
         for b15 being Element of b6
         for b16 being Element of b7
         for b17 being Element of b8
         for b18 being Element of b9
               st b10 = [b11,b12,b13,b14,b15,b16,b17,b18]
            holds b1 = b18)
   holds b1 = b10 `8;

:: MCART_3:th 114
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
      st b1 in [:b2,b3,b4,b5,b6,b7,b8,b9:]
   holds ex b10, b11, b12, b13, b14, b15, b16, b17 being set st
      b10 in b2 & b11 in b3 & b12 in b4 & b13 in b5 & b14 in b6 & b15 in b7 & b16 in b8 & b17 in b9 & b1 = [b10,b11,b12,b13,b14,b15,b16,b17];

:: MCART_3:th 115
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16 being set holds
   [b1,b2,b3,b4,b5,b6,b7,b8] in [:b9,b10,b11,b12,b13,b14,b15,b16:]
iff
   b1 in b9 & b2 in b10 & b3 in b11 & b4 in b12 & b5 in b13 & b6 in b14 & b7 in b15 & b8 in b16;

:: MCART_3:th 116
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
      st for b10 being set holds
              b10 in b9
           iff
              ex b11, b12, b13, b14, b15, b16, b17, b18 being set st
                 b11 in b1 & b12 in b2 & b13 in b3 & b14 in b4 & b15 in b5 & b16 in b6 & b17 in b7 & b18 in b8 & b10 = [b11,b12,b13,b14,b15,b16,b17,b18]
   holds b9 = [:b1,b2,b3,b4,b5,b6,b7,b8:];

:: MCART_3:th 117
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {} & b10 <> {} & b11 <> {} & b12 <> {} & b13 <> {} & b14 <> {} & b15 <> {} & b16 <> {}
for b17 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
for b18 being Element of [:b9,b10,b11,b12,b13,b14,b15,b16:]
      st b17 = b18
   holds b17 `1 = b18 `1 & b17 `2 = b18 `2 & b17 `3 = b18 `3 & b17 `4 = b18 `4 & b17 `5 = b18 `5 & b17 `6 = b18 `6 & b17 `7 = b18 `7 & b17 `8 = b18 `8;

:: MCART_3:th 118
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of bool b1
for b10 being Element of bool b2
for b11 being Element of bool b3
for b12 being Element of bool b4
for b13 being Element of bool b5
for b14 being Element of bool b6
for b15 being Element of bool b7
for b16 being Element of bool b8
for b17 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8:]
      st b17 in [:b9,b10,b11,b12,b13,b14,b15,b16:]
   holds b17 `1 in b9 & b17 `2 in b10 & b17 `3 in b11 & b17 `4 in b12 & b17 `5 in b13 & b17 `6 in b14 & b17 `7 in b15 & b17 `8 in b16;

:: MCART_3:th 119
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16 being set
      st b1 c= b9 & b2 c= b10 & b3 c= b11 & b4 c= b12 & b5 c= b13 & b6 c= b14 & b7 c= b15 & b8 c= b16
   holds [:b1,b2,b3,b4,b5,b6,b7,b8:] c= [:b9,b10,b11,b12,b13,b14,b15,b16:];

:: MCART_3:th 120
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Element of bool b1
for b10 being Element of bool b2
for b11 being Element of bool b3
for b12 being Element of bool b4
for b13 being Element of bool b5
for b14 being Element of bool b6
for b15 being Element of bool b7
for b16 being Element of bool b8 holds
   [:b9,b10,b11,b12,b13,b14,b15,b16:] is Element of bool [:b1,b2,b3,b4,b5,b6,b7,b8:];

:: MCART_3:th 121
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16 being set
             st b3 in b4 & b4 in b5 & b5 in b6 & b6 in b7 & b7 in b8 & b8 in b9 & b9 in b10 & b10 in b11 & b11 in b12 & b12 in b13 & b13 in b14 & b14 in b15 & b15 in b16 & b16 in b2
          holds b3 misses b1);

:: MCART_3:th 122
theorem
for b1 being set
      st b1 <> {}
   holds ex b2 being set st
      b2 in b1 &
       (for b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17 being set
             st b3 in b4 & b4 in b5 & b5 in b6 & b6 in b7 & b7 in b8 & b8 in b9 & b9 in b10 & b10 in b11 & b11 in b12 & b12 in b13 & b13 in b14 & b14 in b15 & b15 in b16 & b16 in b17 & b17 in b2
          holds b3 misses b1);

:: MCART_3:funcnot 28 => MCART_3:func 28
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  func [A1,A2,A3,A4,A5,A6,A7,A8,A9] -> set equals
    [[a1,a2,a3,a4,a5,a6,a7,a8],a9];
end;

:: MCART_3:def 28
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4,b5,b6,b7,b8],b9];

:: MCART_3:th 123
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[[[[[[[b1,b2],b3],b4],b5],b6],b7],b8],b9];

:: MCART_3:th 124
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4,b5,b6,b7],b8,b9];

:: MCART_3:th 125
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4,b5,b6],b7,b8,b9];

:: MCART_3:th 126
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4,b5],b6,b7,b8,b9];

:: MCART_3:th 127
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4],b5,b6,b7,b8,b9];

:: MCART_3:th 128
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3],b4,b5,b6,b7,b8,b9];

:: MCART_3:th 129
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2],b3,b4,b5,b6,b7,b8,b9];

:: MCART_3:th 130
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18 being set
      st [b1,b2,b3,b4,b5,b6,b7,b8,b9] = [b10,b11,b12,b13,b14,b15,b16,b17,b18]
   holds b1 = b10 & b2 = b11 & b3 = b12 & b4 = b13 & b5 = b14 & b6 = b15 & b7 = b16 & b8 = b17 & b9 = b18;

:: MCART_3:funcnot 29 => MCART_3:func 29
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  func [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] -> set equals
    [:[:a1,a2,a3,a4,a5,a6,a7,a8:],a9:];
end;

:: MCART_3:def 29
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4,b5,b6,b7,b8:],b9:];

:: MCART_3:th 131
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:[:[:[:[:[:[:b1,b2:],b3:],b4:],b5:],b6:],b7:],b8:],b9:];

:: MCART_3:th 132
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4,b5,b6,b7:],b8,b9:];

:: MCART_3:th 133
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4,b5,b6:],b7,b8,b9:];

:: MCART_3:th 134
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4,b5:],b6,b7,b8,b9:];

:: MCART_3:th 135
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4:],b5,b6,b7,b8,b9:];

:: MCART_3:th 136
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3:],b4,b5,b6,b7,b8,b9:];

:: MCART_3:th 137
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2:],b3,b4,b5,b6,b7,b8,b9:];

:: MCART_3:th 138
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
   b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
iff
   [:b1,b2,b3,b4,b5,b6,b7,b8,b9:] <> {};

:: MCART_3:th 139
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18 being set
      st b1 <> {} &
         b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         [:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:b10,b11,b12,b13,b14,b15,b16,b17,b18:]
   holds b1 = b10 & b2 = b11 & b3 = b12 & b4 = b13 & b5 = b14 & b6 = b15 & b7 = b16 & b8 = b17 & b9 = b18;

:: MCART_3:th 140
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18 being set
      st [:b1,b2,b3,b4,b5,b6,b7,b8,b9:] <> {} &
         [:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:b10,b11,b12,b13,b14,b15,b16,b17,b18:]
   holds b1 = b10 & b2 = b11 & b3 = b12 & b4 = b13 & b5 = b14 & b6 = b15 & b7 = b16 & b8 = b17 & b9 = b18;

:: MCART_3:th 141
theorem
for b1, b2 being set
      st [:b1,b1,b1,b1,b1,b1,b1,b1,b1:] = [:b2,b2,b2,b2,b2,b2,b2,b2,b2:]
   holds b1 = b2;

:: MCART_3:th 142
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
   ex b11 being Element of b1 st
      ex b12 being Element of b2 st
         ex b13 being Element of b3 st
            ex b14 being Element of b4 st
               ex b15 being Element of b5 st
                  ex b16 being Element of b6 st
                     ex b17 being Element of b7 st
                        ex b18 being Element of b8 st
                           ex b19 being Element of b9 st
                              b10 = [b11,b12,b13,b14,b15,b16,b17,b18,b19];

:: MCART_3:funcnot 30 => MCART_3:func 30
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  let a10 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8,a9:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {} & a9 <> {};
  func A10 `1 -> Element of a1 means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
          st a10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9]
       holds it = b1;
end;

:: MCART_3:def 30
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11 being Element of b1 holds
      b11 = b10 `1
   iff
      for b12, b13, b14, b15, b16, b17, b18, b19, b20 being set
            st b10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
         holds b11 = b12;

:: MCART_3:funcnot 31 => MCART_3:func 31
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  let a10 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8,a9:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {} & a9 <> {};
  func A10 `2 -> Element of a2 means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
          st a10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9]
       holds it = b2;
end;

:: MCART_3:def 31
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11 being Element of b2 holds
      b11 = b10 `2
   iff
      for b12, b13, b14, b15, b16, b17, b18, b19, b20 being set
            st b10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
         holds b11 = b13;

:: MCART_3:funcnot 32 => MCART_3:func 32
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  let a10 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8,a9:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {} & a9 <> {};
  func A10 `3 -> Element of a3 means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
          st a10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9]
       holds it = b3;
end;

:: MCART_3:def 32
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11 being Element of b3 holds
      b11 = b10 `3
   iff
      for b12, b13, b14, b15, b16, b17, b18, b19, b20 being set
            st b10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
         holds b11 = b14;

:: MCART_3:funcnot 33 => MCART_3:func 33
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  let a10 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8,a9:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {} & a9 <> {};
  func A10 `4 -> Element of a4 means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
          st a10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9]
       holds it = b4;
end;

:: MCART_3:def 33
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11 being Element of b4 holds
      b11 = b10 `4
   iff
      for b12, b13, b14, b15, b16, b17, b18, b19, b20 being set
            st b10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
         holds b11 = b15;

:: MCART_3:funcnot 34 => MCART_3:func 34
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  let a10 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8,a9:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {} & a9 <> {};
  func A10 `5 -> Element of a5 means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
          st a10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9]
       holds it = b5;
end;

:: MCART_3:def 34
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11 being Element of b5 holds
      b11 = b10 `5
   iff
      for b12, b13, b14, b15, b16, b17, b18, b19, b20 being set
            st b10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
         holds b11 = b16;

:: MCART_3:funcnot 35 => MCART_3:func 35
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  let a10 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8,a9:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {} & a9 <> {};
  func A10 `6 -> Element of a6 means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
          st a10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9]
       holds it = b6;
end;

:: MCART_3:def 35
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11 being Element of b6 holds
      b11 = b10 `6
   iff
      for b12, b13, b14, b15, b16, b17, b18, b19, b20 being set
            st b10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
         holds b11 = b17;

:: MCART_3:funcnot 36 => MCART_3:func 36
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  let a10 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8,a9:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {} & a9 <> {};
  func A10 `7 -> Element of a7 means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
          st a10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9]
       holds it = b7;
end;

:: MCART_3:def 36
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11 being Element of b7 holds
      b11 = b10 `7
   iff
      for b12, b13, b14, b15, b16, b17, b18, b19, b20 being set
            st b10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
         holds b11 = b18;

:: MCART_3:funcnot 37 => MCART_3:func 37
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  let a10 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8,a9:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {} & a9 <> {};
  func A10 `8 -> Element of a8 means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
          st a10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9]
       holds it = b8;
end;

:: MCART_3:def 37
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11 being Element of b8 holds
      b11 = b10 `8
   iff
      for b12, b13, b14, b15, b16, b17, b18, b19, b20 being set
            st b10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
         holds b11 = b19;

:: MCART_3:funcnot 38 => MCART_3:func 38
definition
  let a1, a2, a3, a4, a5, a6, a7, a8, a9 be set;
  let a10 be Element of [:a1,a2,a3,a4,a5,a6,a7,a8,a9:];
  assume a1 <> {} & a2 <> {} & a3 <> {} & a4 <> {} & a5 <> {} & a6 <> {} & a7 <> {} & a8 <> {} & a9 <> {};
  func A10 `9 -> Element of a9 means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
          st a10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9]
       holds it = b9;
end;

:: MCART_3:def 38
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11 being Element of b9 holds
      b11 = b10 `9
   iff
      for b12, b13, b14, b15, b16, b17, b18, b19, b20 being set
            st b10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
         holds b11 = b20;

:: MCART_3:th 143
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b11, b12, b13, b14, b15, b16, b17, b18, b19 being set
      st b10 = [b11,b12,b13,b14,b15,b16,b17,b18,b19]
   holds b10 `1 = b11 & b10 `2 = b12 & b10 `3 = b13 & b10 `4 = b14 & b10 `5 = b15 & b10 `6 = b16 & b10 `7 = b17 & b10 `8 = b18 & b10 `9 = b19;

:: MCART_3:th 144
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
   b10 = [b10 `1,b10 `2,b10 `3,b10 `4,b10 `5,b10 `6,b10 `7,b10 `8,b10 `9];

:: MCART_3:th 145
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
   b10 `1 = b10 `1 `1 `1 `1 `1 `1 `1 `1 &
    b10 `2 = b10 `1 `1 `1 `1 `1 `1 `1 `2 &
    b10 `3 = b10 `1 `1 `1 `1 `1 `1 `2 &
    b10 `4 = b10 `1 `1 `1 `1 `1 `2 &
    b10 `5 = b10 `1 `1 `1 `1 `2 &
    b10 `6 = b10 `1 `1 `1 `2 &
    b10 `7 = b10 `1 `1 `2 &
    b10 `8 = b10 `1 `2 &
    b10 `9 = b10 `2;

:: MCART_3:th 146
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18 being set
      st [:b1,b2,b3,b4,b5,b6,b7,b8,b9:] meets [:b10,b11,b12,b13,b14,b15,b16,b17,b18:]
   holds b1 meets b10 & b2 meets b11 & b3 meets b12 & b4 meets b13 & b5 meets b14 & b6 meets b15 & b7 meets b16 & b8 meets b17 & b9 meets b18;

:: MCART_3:th 147
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds
[:{b1},{b2},{b3},{b4},{b5},{b6},{b7},{b8},{b9}:] = {[b1,b2,b3,b4,b5,b6,b7,b8,b9]};

:: MCART_3:th 148
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {}
for b11, b12, b13, b14, b15, b16, b17, b18, b19 being set
      st b10 = [b11,b12,b13,b14,b15,b16,b17,b18,b19]
   holds b10 `1 = b11 & b10 `2 = b12 & b10 `3 = b13 & b10 `4 = b14 & b10 `5 = b15 & b10 `6 = b16 & b10 `7 = b17 & b10 `8 = b18 & b10 `9 = b19;

:: MCART_3:th 149
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
for b11 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         b10 <> {} &
         (for b12 being Element of b2
         for b13 being Element of b3
         for b14 being Element of b4
         for b15 being Element of b5
         for b16 being Element of b6
         for b17 being Element of b7
         for b18 being Element of b8
         for b19 being Element of b9
         for b20 being Element of b10
               st b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
            holds b1 = b12)
   holds b1 = b11 `1;

:: MCART_3:th 150
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
for b11 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         b10 <> {} &
         (for b12 being Element of b2
         for b13 being Element of b3
         for b14 being Element of b4
         for b15 being Element of b5
         for b16 being Element of b6
         for b17 being Element of b7
         for b18 being Element of b8
         for b19 being Element of b9
         for b20 being Element of b10
               st b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
            holds b1 = b13)
   holds b1 = b11 `2;

:: MCART_3:th 151
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
for b11 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         b10 <> {} &
         (for b12 being Element of b2
         for b13 being Element of b3
         for b14 being Element of b4
         for b15 being Element of b5
         for b16 being Element of b6
         for b17 being Element of b7
         for b18 being Element of b8
         for b19 being Element of b9
         for b20 being Element of b10
               st b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
            holds b1 = b14)
   holds b1 = b11 `3;

:: MCART_3:th 152
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
for b11 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         b10 <> {} &
         (for b12 being Element of b2
         for b13 being Element of b3
         for b14 being Element of b4
         for b15 being Element of b5
         for b16 being Element of b6
         for b17 being Element of b7
         for b18 being Element of b8
         for b19 being Element of b9
         for b20 being Element of b10
               st b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
            holds b1 = b15)
   holds b1 = b11 `4;

:: MCART_3:th 153
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
for b11 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         b10 <> {} &
         (for b12 being Element of b2
         for b13 being Element of b3
         for b14 being Element of b4
         for b15 being Element of b5
         for b16 being Element of b6
         for b17 being Element of b7
         for b18 being Element of b8
         for b19 being Element of b9
         for b20 being Element of b10
               st b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
            holds b1 = b16)
   holds b1 = b11 `5;

:: MCART_3:th 154
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
for b11 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         b10 <> {} &
         (for b12 being Element of b2
         for b13 being Element of b3
         for b14 being Element of b4
         for b15 being Element of b5
         for b16 being Element of b6
         for b17 being Element of b7
         for b18 being Element of b8
         for b19 being Element of b9
         for b20 being Element of b10
               st b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
            holds b1 = b17)
   holds b1 = b11 `6;

:: MCART_3:th 155
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
for b11 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         b10 <> {} &
         (for b12 being Element of b2
         for b13 being Element of b3
         for b14 being Element of b4
         for b15 being Element of b5
         for b16 being Element of b6
         for b17 being Element of b7
         for b18 being Element of b8
         for b19 being Element of b9
         for b20 being Element of b10
               st b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
            holds b1 = b18)
   holds b1 = b11 `7;

:: MCART_3:th 156
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
for b11 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         b10 <> {} &
         (for b12 being Element of b2
         for b13 being Element of b3
         for b14 being Element of b4
         for b15 being Element of b5
         for b16 being Element of b6
         for b17 being Element of b7
         for b18 being Element of b8
         for b19 being Element of b9
         for b20 being Element of b10
               st b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
            holds b1 = b19)
   holds b1 = b11 `8;

:: MCART_3:th 157
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
for b11 being Element of [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
      st b2 <> {} &
         b3 <> {} &
         b4 <> {} &
         b5 <> {} &
         b6 <> {} &
         b7 <> {} &
         b8 <> {} &
         b9 <> {} &
         b10 <> {} &
         (for b12 being Element of b2
         for b13 being Element of b3
         for b14 being Element of b4
         for b15 being Element of b5
         for b16 being Element of b6
         for b17 being Element of b7
         for b18 being Element of b8
         for b19 being Element of b9
         for b20 being Element of b10
               st b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
            holds b1 = b20)
   holds b1 = b11 `9;

:: MCART_3:th 158
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
      st b1 in [:b2,b3,b4,b5,b6,b7,b8,b9,b10:]
   holds ex b11, b12, b13, b14, b15, b16, b17, b18, b19 being set st
      b11 in b2 & b12 in b3 & b13 in b4 & b14 in b5 & b15 in b6 & b16 in b7 & b17 in b8 & b18 in b9 & b19 in b10 & b1 = [b11,b12,b13,b14,b15,b16,b17,b18,b19];

:: MCART_3:th 159
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18 being set holds
   [b1,b2,b3,b4,b5,b6,b7,b8,b9] in [:b10,b11,b12,b13,b14,b15,b16,b17,b18:]
iff
   b1 in b10 & b2 in b11 & b3 in b12 & b4 in b13 & b5 in b14 & b6 in b15 & b7 in b16 & b8 in b17 & b9 in b18;

:: MCART_3:th 160
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being set
      st for b11 being set holds
              b11 in b10
           iff
              ex b12, b13, b14, b15, b16, b17, b18, b19, b20 being set st
                 b12 in b1 & b13 in b2 & b14 in b3 & b15 in b4 & b16 in b5 & b17 in b6 & b18 in b7 & b19 in b8 & b20 in b9 & b11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20]
   holds b10 = [:b1,b2,b3,b4,b5,b6,b7,b8,b9:];

:: MCART_3:th 161
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18 being set
   st b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} & b5 <> {} & b6 <> {} & b7 <> {} & b8 <> {} & b9 <> {} & b10 <> {} & b11 <> {} & b12 <> {} & b13 <> {} & b14 <> {} & b15 <> {} & b16 <> {} & b17 <> {} & b18 <> {}
for b19 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
for b20 being Element of [:b10,b11,b12,b13,b14,b15,b16,b17,b18:]
      st b19 = b20
   holds b19 `1 = b20 `1 & b19 `2 = b20 `2 & b19 `3 = b20 `3 & b19 `4 = b20 `4 & b19 `5 = b20 `5 & b19 `6 = b20 `6 & b19 `7 = b20 `7 & b19 `8 = b20 `8 & b19 `9 = b20 `9;

:: MCART_3:th 162
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of bool b1
for b11 being Element of bool b2
for b12 being Element of bool b3
for b13 being Element of bool b4
for b14 being Element of bool b5
for b15 being Element of bool b6
for b16 being Element of bool b7
for b17 being Element of bool b8
for b18 being Element of bool b9
for b19 being Element of [:b1,b2,b3,b4,b5,b6,b7,b8,b9:]
      st b19 in [:b10,b11,b12,b13,b14,b15,b16,b17,b18:]
   holds b19 `1 in b10 & b19 `2 in b11 & b19 `3 in b12 & b19 `4 in b13 & b19 `5 in b14 & b19 `6 in b15 & b19 `7 in b16 & b19 `8 in b17 & b19 `9 in b18;

:: MCART_3:th 163
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18 being set
      st b1 c= b10 & b2 c= b11 & b3 c= b12 & b4 c= b13 & b5 c= b14 & b6 c= b15 & b7 c= b16 & b8 c= b17 & b9 c= b18
   holds [:b1,b2,b3,b4,b5,b6,b7,b8,b9:] c= [:b10,b11,b12,b13,b14,b15,b16,b17,b18:];

:: MCART_3:th 164
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Element of bool b1
for b11 being Element of bool b2
for b12 being Element of bool b3
for b13 being Element of bool b4
for b14 being Element of bool b5
for b15 being Element of bool b6
for b16 being Element of bool b7
for b17 being Element of bool b8
for b18 being Element of bool b9 holds
   [:b10,b11,b12,b13,b14,b15,b16,b17,b18:] is Element of bool [:b1,b2,b3,b4,b5,b6,b7,b8,b9:];