Article AFVECT01, MML version 4.99.1005

:: AFVECT01:funcreg 1
registration
  let a1 be non empty set;
  let a2 be Relation of [:a1,a1:],[:a1,a1:];
  cluster AffinStruct(#a1,a2#) -> non empty strict;
end;

:: AFVECT01:attrnot 1 => AFVECT01:attr 1
definition
  let a1 be non empty AffinStruct;
  attr a1 is WeakAffSegm-like means
    (for b1, b2 being Element of the carrier of a1 holds
     b1,b2 // b2,b1) &
     (for b1, b2 being Element of the carrier of a1
           st b1,b2 // b1,b1
        holds b1 = b2) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b5,b6 & b3,b4 // b5,b6
        holds b1,b2 // b3,b4) &
     (for b1, b2 being Element of the carrier of a1 holds
     ex b3 being Element of the carrier of a1 st
        b1,b3 // b3,b2) &
     (for b1, b2, b3, b4, b5 being Element of the carrier of a1
           st b1 <> b2 & b3 <> b4 & b5,b1 // b5,b2 & b5,b3 // b5,b4
        holds b1,b3 // b2,b4) &
     (for b1, b2 being Element of the carrier of a1
           st b1 <> b2
        holds ex b3 being Element of the carrier of a1 st
           ((b1 <> b3 implies not b1,b2 // b2,b3) implies ex b4, b5 being Element of the carrier of a1 st
              b4 <> b5 & b1,b2 // b4,b5 & b1,b4 // b4,b2 & b1,b5 // b5,b2)) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b2,b6 & b2,b3 // b4,b5 & b2,b4 // b4,b3 & b2,b5 // b5,b3
        holds b1,b3 // b3,b6) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b1 <> b4 & b2 <> b3 & b1,b2 // b2,b4 & b1,b3 // b3,b4
        holds ex b5, b6 being Element of the carrier of a1 st
           b5 <> b6 & b2,b3 // b5,b6 & b2,b5 // b5,b3 & b2,b6 // b6,b3) &
     (for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
           st b1,b2 // b4,b5 & b1,b3 // b6,b7 & b1,b4 // b4,b2 & b1,b6 // b6,b3 & b1,b5 // b5,b2 & b1,b7 // b7,b3
        holds ex b8, b9 being Element of the carrier of a1 st
           b2,b3 // b8,b9 & b2,b8 // b8,b3 & b2,b9 // b9,b3);
end;

:: AFVECT01:dfs 1
definiens
  let a1 be non empty AffinStruct;
To prove
     a1 is WeakAffSegm-like
it is sufficient to prove
  thus (for b1, b2 being Element of the carrier of a1 holds
     b1,b2 // b2,b1) &
     (for b1, b2 being Element of the carrier of a1
           st b1,b2 // b1,b1
        holds b1 = b2) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b5,b6 & b3,b4 // b5,b6
        holds b1,b2 // b3,b4) &
     (for b1, b2 being Element of the carrier of a1 holds
     ex b3 being Element of the carrier of a1 st
        b1,b3 // b3,b2) &
     (for b1, b2, b3, b4, b5 being Element of the carrier of a1
           st b1 <> b2 & b3 <> b4 & b5,b1 // b5,b2 & b5,b3 // b5,b4
        holds b1,b3 // b2,b4) &
     (for b1, b2 being Element of the carrier of a1
           st b1 <> b2
        holds ex b3 being Element of the carrier of a1 st
           ((b1 <> b3 implies not b1,b2 // b2,b3) implies ex b4, b5 being Element of the carrier of a1 st
              b4 <> b5 & b1,b2 // b4,b5 & b1,b4 // b4,b2 & b1,b5 // b5,b2)) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b2,b6 & b2,b3 // b4,b5 & b2,b4 // b4,b3 & b2,b5 // b5,b3
        holds b1,b3 // b3,b6) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b1 <> b4 & b2 <> b3 & b1,b2 // b2,b4 & b1,b3 // b3,b4
        holds ex b5, b6 being Element of the carrier of a1 st
           b5 <> b6 & b2,b3 // b5,b6 & b2,b5 // b5,b3 & b2,b6 // b6,b3) &
     (for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
           st b1,b2 // b4,b5 & b1,b3 // b6,b7 & b1,b4 // b4,b2 & b1,b6 // b6,b3 & b1,b5 // b5,b2 & b1,b7 // b7,b3
        holds ex b8, b9 being Element of the carrier of a1 st
           b2,b3 // b8,b9 & b2,b8 // b8,b3 & b2,b9 // b9,b3);

:: AFVECT01:def 2
theorem
for b1 being non empty AffinStruct holds
      b1 is WeakAffSegm-like
   iff
      (for b2, b3 being Element of the carrier of b1 holds
       b2,b3 // b3,b2) &
       (for b2, b3 being Element of the carrier of b1
             st b2,b3 // b2,b2
          holds b2 = b3) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
             st b2,b3 // b6,b7 & b4,b5 // b6,b7
          holds b2,b3 // b4,b5) &
       (for b2, b3 being Element of the carrier of b1 holds
       ex b4 being Element of the carrier of b1 st
          b2,b4 // b4,b3) &
       (for b2, b3, b4, b5, b6 being Element of the carrier of b1
             st b2 <> b3 & b4 <> b5 & b6,b2 // b6,b3 & b6,b4 // b6,b5
          holds b2,b4 // b3,b5) &
       (for b2, b3 being Element of the carrier of b1
             st b2 <> b3
          holds ex b4 being Element of the carrier of b1 st
             ((b2 <> b4 implies not b2,b3 // b3,b4) implies ex b5, b6 being Element of the carrier of b1 st
                b5 <> b6 & b2,b3 // b5,b6 & b2,b5 // b5,b3 & b2,b6 // b6,b3)) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
             st b2,b3 // b3,b7 & b3,b4 // b5,b6 & b3,b5 // b5,b4 & b3,b6 // b6,b4
          holds b2,b4 // b4,b7) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b2 <> b5 & b3 <> b4 & b2,b3 // b3,b5 & b2,b4 // b4,b5
          holds ex b6, b7 being Element of the carrier of b1 st
             b6 <> b7 & b3,b4 // b6,b7 & b3,b6 // b6,b4 & b3,b7 // b7,b4) &
       (for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
             st b2,b3 // b5,b6 & b2,b4 // b7,b8 & b2,b5 // b5,b3 & b2,b7 // b7,b4 & b2,b6 // b6,b3 & b2,b8 // b8,b4
          holds ex b9, b10 being Element of the carrier of b1 st
             b3,b4 // b9,b10 & b3,b9 // b9,b4 & b3,b10 // b10,b4);

:: AFVECT01:exreg 1
registration
  cluster non empty non trivial strict WeakAffSegm-like AffinStruct;
end;

:: AFVECT01:modenot 1
definition
  mode WeakAffSegm is non empty non trivial WeakAffSegm-like AffinStruct;
end;

:: AFVECT01:th 1
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 // b2,b3;

:: AFVECT01:th 2
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b4,b5 // b2,b3;

:: AFVECT01:th 3
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b2,b3 // b5,b4;

:: AFVECT01:th 4
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b3,b2 // b4,b5;

:: AFVECT01:th 5
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b2 // b3,b3;

:: AFVECT01:th 6
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3 // b4,b4
   holds b2 = b3;

:: AFVECT01:th 7
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2,b3 // b4,b5 & b2,b3 // b3,b6 & b2,b4 // b4,b3 & b2,b5 // b5,b3
   holds b2 = b6;

:: AFVECT01:th 8
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b2,b4 & b2,b5 // b2,b4 & b5 <> b3 & b5 <> b4
   holds b3 = b4;

:: AFVECT01:prednot 1 => AFVECT01:pred 1
definition
  let a1 be non empty non trivial WeakAffSegm-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  pred MDist A2,A3 means
    ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a2,a3 // b1,b2 & a2,b1 // b1,a3 & a2,b2 // b2,a3;
end;

:: AFVECT01:dfs 2
definiens
  let a1 be non empty non trivial WeakAffSegm-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
To prove
     MDist a2,a3
it is sufficient to prove
  thus ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a2,a3 // b1,b2 & a2,b1 // b1,a3 & a2,b2 // b2,a3;

:: AFVECT01:def 4
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
   MDist b2,b3
iff
   ex b4, b5 being Element of the carrier of b1 st
      b4 <> b5 & b2,b3 // b4,b5 & b2,b4 // b4,b3 & b2,b5 // b5,b3;

:: AFVECT01:prednot 2 => AFVECT01:pred 2
definition
  let a1 be non empty non trivial WeakAffSegm-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  pred Mid A2,A3,A4 means
    ((a2 = a3 & a3 = a4 implies a2 <> a4) & (a2 = a4 implies not MDist a2,a3)) implies a2 <> a4 & a2,a3 // a3,a4;
end;

:: AFVECT01:dfs 3
definiens
  let a1 be non empty non trivial WeakAffSegm-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
To prove
     Mid a2,a3,a4
it is sufficient to prove
  thus ((a2 = a3 & a3 = a4 implies a2 <> a4) & (a2 = a4 implies not MDist a2,a3)) implies a2 <> a4 & a2,a3 // a3,a4;

:: AFVECT01:def 5
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   Mid b2,b3,b4
iff
   ((b2 = b3 & b3 = b4 implies b2 <> b4) & (b2 = b4 implies not MDist b2,b3) implies b2 <> b4 & b2,b3 // b3,b4);

:: AFVECT01:th 11
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st b2 <> b3 & not MDist b2,b3
   holds ex b4 being Element of the carrier of b1 st
      b2 <> b4 & b2,b3 // b3,b4;

:: AFVECT01:th 12
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st MDist b2,b3 & b2,b3 // b3,b4
   holds b2 = b4;

:: AFVECT01:th 13
theorem
for b1 being non empty non trivial WeakAffSegm-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st MDist b2,b3
   holds b2 <> b3;