Article AFF_1, MML version 4.99.1005

:: AFF_1:prednot 1 => AFF_1:pred 1
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  pred LIN A2,A3,A4 means
    a2,a3 // a2,a4;
end;

:: AFF_1:dfs 1
definiens
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
To prove
     LIN a2,a3,a4
it is sufficient to prove
  thus a2,a3 // a2,a4;

:: AFF_1:def 1
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   LIN b2,b3,b4
iff
   b2,b3 // b2,b4;

:: AFF_1:th 10
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1 holds
   ex b3 being Element of the carrier of b1 st
      b2 <> b3;

:: AFF_1:th 11
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 // b3,b2 & b2,b3 // b2,b3;

:: AFF_1:th 12
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 // b4,b4 & b4,b4 // b2,b3;

:: AFF_1:th 13
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b2,b3 // b5,b4 & b3,b2 // b4,b5 & b3,b2 // b5,b4 & b4,b5 // b2,b3 & b4,b5 // b3,b2 & b5,b4 // b2,b3 & b5,b4 // b3,b2;

:: AFF_1:th 14
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2 <> b3 &
         ((b2,b3 // b4,b5 implies not b2,b3 // b6,b7) & (b2,b3 // b4,b5 implies not b6,b7 // b2,b3) & (b4,b5 // b2,b3 implies not b6,b7 // b2,b3) implies b4,b5 // b2,b3 & b2,b3 // b6,b7)
   holds b4,b5 // b6,b7;

:: AFF_1:th 15
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st LIN b2,b3,b4
   holds LIN b2,b4,b3 & LIN b3,b2,b4 & LIN b3,b4,b2 & LIN b4,b2,b3 & LIN b4,b3,b2;

:: AFF_1:th 16
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
LIN b2,b2,b3 & LIN b2,b3,b3 & LIN b2,b3,b2;

:: AFF_1:th 17
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2 <> b3 & LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b3,b6
   holds LIN b4,b5,b6;

:: AFF_1:th 18
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <> b3 & LIN b2,b3,b4 & b2,b3 // b4,b5
   holds LIN b2,b3,b5;

:: AFF_1:th 19
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st LIN b2,b3,b4 & LIN b2,b3,b5
   holds b2,b3 // b4,b5;

:: AFF_1:th 20
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2 <> b3 & LIN b4,b5,b2 & LIN b4,b5,b3 & LIN b2,b3,b6
   holds LIN b4,b5,b6;

:: AFF_1:th 21
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
   ex b2, b3, b4 being Element of the carrier of b1 st
      not LIN b2,b3,b4;

:: AFF_1:th 22
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st b2 <> b3
   holds ex b4 being Element of the carrier of b1 st
      not LIN b2,b3,b4;

:: AFF_1:th 23
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not LIN b2,b3,b4 & LIN b2,b4,b5 & b3,b4 // b3,b5
   holds b4 = b5;

:: AFF_1:funcnot 1 => AFF_1:func 1
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  func Line(A2,A3) -> Element of bool the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
          b1 in it
       iff
          LIN a2,a3,b1;
end;

:: AFF_1:def 2
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
      b4 = Line(b2,b3)
   iff
      for b5 being Element of the carrier of b1 holds
            b5 in b4
         iff
            LIN b2,b3,b5;

:: AFF_1:th 25
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
Line(b2,b3) = Line(b3,b2);

:: AFF_1:funcnot 2 => AFF_1:func 2
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  redefine func Line(a2,a3) -> Element of bool the carrier of a1;
  commutativity;
::  for a1 being non empty non trivial AffinSpace-like AffinStruct
::  for a2, a3 being Element of the carrier of a1 holds
::  Line(a2,a3) = Line(a3,a2);
end;

:: AFF_1:th 26
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2 in Line(b2,b3) & b3 in Line(b2,b3);

:: AFF_1:th 27
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 in Line(b3,b4) & b5 in Line(b3,b4) & b2 <> b5
   holds Line(b2,b5) c= Line(b3,b4);

:: AFF_1:th 28
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 in Line(b3,b4) & b5 in Line(b3,b4) & b3 <> b4
   holds Line(b3,b4) c= Line(b2,b5);

:: AFF_1:attrnot 1 => AFF_1:attr 1
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is being_line means
    ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a2 = Line(b1,b2);
end;

:: AFF_1:dfs 3
definiens
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is being_line
it is sufficient to prove
  thus ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a2 = Line(b1,b2);

:: AFF_1:def 3
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is being_line(b1)
   iff
      ex b3, b4 being Element of the carrier of b1 st
         b3 <> b4 & b2 = Line(b3,b4);

:: AFF_1:prednot 2 => AFF_1:attr 1
notation
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Element of bool the carrier of a1;
  synonym a2 is_line for being_line;
end;

:: AFF_1:th 30
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
      st b4 is being_line(b1) & b5 is being_line(b1) & b2 in b4 & b3 in b4 & b2 in b5 & b3 in b5 & b2 <> b3
   holds b4 = b5;

:: AFF_1:th 31
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1
      st b2 is being_line(b1)
   holds ex b3, b4 being Element of the carrier of b1 st
      b3 in b2 & b4 in b2 & b3 <> b4;

:: AFF_1:th 32
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 is being_line(b1)
   holds ex b4 being Element of the carrier of b1 st
      b2 <> b4 & b4 in b3;

:: AFF_1:th 33
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   LIN b2,b3,b4
iff
   ex b5 being Element of bool the carrier of b1 st
      b5 is being_line(b1) & b2 in b5 & b3 in b5 & b4 in b5;

:: AFF_1:prednot 3 => AFF_1:pred 2
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  let a4 be Element of bool the carrier of a1;
  pred A2,A3 // A4 means
    ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a4 = Line(b1,b2) & a2,a3 // b1,b2;
end;

:: AFF_1:dfs 4
definiens
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  let a4 be Element of bool the carrier of a1;
To prove
     a2,a3 // a4
it is sufficient to prove
  thus ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a4 = Line(b1,b2) & a2,a3 // b1,b2;

:: AFF_1:def 4
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
      b2,b3 // b4
   iff
      ex b5, b6 being Element of the carrier of b1 st
         b5 <> b6 & b4 = Line(b5,b6) & b2,b3 // b5,b6;

:: AFF_1:prednot 4 => AFF_1:pred 3
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2, a3 be Element of bool the carrier of a1;
  pred A2 // A3 means
    ex b1, b2 being Element of the carrier of a1 st
       a2 = Line(b1,b2) & b1 <> b2 & b1,b2 // a3;
end;

:: AFF_1:dfs 5
definiens
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2, a3 be Element of bool the carrier of a1;
To prove
     a2 // a3
it is sufficient to prove
  thus ex b1, b2 being Element of the carrier of a1 st
       a2 = Line(b1,b2) & b1 <> b2 & b1,b2 // a3;

:: AFF_1:def 5
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2 // b3
iff
   ex b4, b5 being Element of the carrier of b1 st
      b2 = Line(b4,b5) & b4 <> b5 & b4,b5 // b3;

:: AFF_1:th 36
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 in Line(b3,b4) & b3 <> b4
   holds    b5 in Line(b3,b4)
   iff
      b3,b4 // b2,b5;

:: AFF_1:th 37
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
      st b4 is being_line(b1) & b2 in b4
   holds    b3 in b4
   iff
      b2,b3 // b4;

:: AFF_1:th 38
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
      b2 <> b3 & b4 = Line(b2,b3)
   iff
      b4 is being_line(b1) & b2 in b4 & b3 in b4 & b2 <> b3;

:: AFF_1:th 39
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of bool the carrier of b1
      st b5 is being_line(b1) & b2 in b5 & b3 in b5 & b2 <> b3 & LIN b2,b3,b4
   holds b4 in b5;

:: AFF_1:th 40
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1
      st ex b3, b4 being Element of the carrier of b1 st
           b3,b4 // b2
   holds b2 is being_line(b1);

:: AFF_1:th 41
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
      st b2 in b6 & b3 in b6 & b6 is being_line(b1) & b2 <> b3
   holds    b4,b5 // b6
   iff
      b4,b5 // b2,b3;

:: AFF_1:th 43
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st b2 <> b3
   holds b2,b3 // Line(b2,b3);

:: AFF_1:th 44
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
      st b4 is being_line(b1)
   holds    b2,b3 // b4
   iff
      ex b5, b6 being Element of the carrier of b1 st
         b5 <> b6 & b5 in b4 & b6 in b4 & b2,b3 // b5,b6;

:: AFF_1:th 45
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
      st b6 is being_line(b1) & b2,b3 // b6 & b4,b5 // b6
   holds b2,b3 // b4,b5;

:: AFF_1:th 46
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
      st b2,b3 // b6 & b2,b3 // b4,b5 & b2 <> b3
   holds b4,b5 // b6;

:: AFF_1:th 47
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 is being_line(b1)
   holds b2,b2 // b3;

:: AFF_1:th 48
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
      st b2,b3 // b4
   holds b3,b2 // b4;

:: AFF_1:th 49
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
      st b2,b3 // b4 & not b2 in b4
   holds not b3 in b4;

:: AFF_1:th 50
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 // b3
   holds b2 is being_line(b1) & b3 is being_line(b1);

:: AFF_1:th 51
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2 // b3
iff
   ex b4, b5, b6, b7 being Element of the carrier of b1 st
      b4 <> b5 & b6 <> b7 & b4,b5 // b6,b7 & b2 = Line(b4,b5) & b3 = Line(b6,b7);

:: AFF_1:th 52
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7 being Element of bool the carrier of b1
      st b6 is being_line(b1) & b7 is being_line(b1) & b2 in b6 & b3 in b6 & b4 in b7 & b5 in b7 & b2 <> b3 & b4 <> b5
   holds    b6 // b7
   iff
      b2,b3 // b4,b5;

:: AFF_1:th 53
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7 being Element of bool the carrier of b1
      st b2 in b6 & b3 in b6 & b4 in b7 & b5 in b7 & b6 // b7
   holds b2,b3 // b4,b5;

:: AFF_1:th 54
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
      st b2 in b4 & b3 in b4 & b4 // b5
   holds b2,b3 // b5;

:: AFF_1:th 55
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1
      st b2 is being_line(b1)
   holds b2 // b2;

:: AFF_1:th 56
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 // b3
   holds b3 // b2;

:: AFF_1:prednot 5 => AFF_1:pred 4
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2, a3 be Element of bool the carrier of a1;
  redefine pred a2 // a3;
  symmetry;
::  for a1 being non empty non trivial AffinSpace-like AffinStruct
::  for a2, a3 being Element of bool the carrier of a1
::        st a2 // a3
::     holds a3 // a2;
end;

:: AFF_1:th 57
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
      st b2,b3 // b4 & b4 // b5
   holds b2,b3 // b5;

:: AFF_1:th 58
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st ((b2 // b3 implies not b3 // b4) & (b2 // b3 implies not b4 // b3) & (b3 // b2 implies not b3 // b4) implies b3 // b2 & b4 // b3)
   holds b2 // b4;

:: AFF_1:th 59
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
      st b3 // b4 & b2 in b3 & b2 in b4
   holds b3 = b4;

:: AFF_1:th 60
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of bool the carrier of b1
      st b2 in b5 & not b3 in b5 & b3,b4 // b5 & b3 <> b4
   holds not LIN b2,b3,b4;

:: AFF_1:th 61
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
for b7 being Element of bool the carrier of b1
      st b2,b3 // b7 & b4,b5 // b7 & LIN b6,b2,b4 & LIN b6,b3,b5 & b6 in b7 & not b2 in b7 & b2 = b3
   holds b4 = b5;

:: AFF_1:th 62
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
      st b6 is being_line(b1) & b2 in b6 & b3 in b6 & b4 in b6 & b2 <> b3 & b2,b3 // b4,b5
   holds b5 in b6;

:: AFF_1:th 63
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 is being_line(b1)
   holds ex b4 being Element of bool the carrier of b1 st
      b2 in b4 & b3 // b4;

:: AFF_1:th 64
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4, b5 being Element of bool the carrier of b1
      st b3 // b4 & b3 // b5 & b2 in b4 & b2 in b5
   holds b4 = b5;

:: AFF_1:th 65
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
      st b6 is being_line(b1) & b2 in b6 & b3 in b6 & b4 in b6 & b5 in b6
   holds b2,b3 // b4,b5;

:: AFF_1:th 66
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
      st b4 is being_line(b1) & b2 in b4 & b3 in b4
   holds b2,b3 // b4;

:: AFF_1:th 67
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
      st b2,b3 // b4 & b2,b3 // b5 & b2 <> b3
   holds b4 // b5;

:: AFF_1:th 68
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b4,b6 & b3,b4 // b5,b6 & b5 = b6
   holds b5 = b2 & b6 = b2;

:: AFF_1:th 69
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b4,b6 & b3,b4 // b5,b6 & b5 = b2
   holds b6 = b2;

:: AFF_1:th 70
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b4,b6 & LIN b2,b4,b7 & b3,b4 // b5,b6 & b3,b4 // b5,b7
   holds b6 = b7;

:: AFF_1:th 71
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
      st b4 is being_line(b1) & b2 in b4 & b3 in b4 & b2 <> b3
   holds b4 = Line(b2,b3);

:: AFF_1:th 72
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is being_line(b1) & b3 is being_line(b1) & not b2 // b3
   holds ex b4 being Element of the carrier of b1 st
      b4 in b2 & b4 in b3;

:: AFF_1:th 73
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
      st b4 is being_line(b1) & not b2,b3 // b4
   holds ex b5 being Element of the carrier of b1 st
      b5 in b4 & LIN b2,b3,b5;

:: AFF_1:th 74
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3 // b4,b5
   holds ex b6 being Element of the carrier of b1 st
      LIN b2,b3,b6 & LIN b4,b5,b6;