Article SEMI_AF1, MML version 4.99.1005

:: SEMI_AF1:attrnot 1 => SEMI_AF1:attr 1
definition
  let a1 be non empty AffinStruct;
  attr a1 is Semi_Affine_Space-like means
    (for b1, b2 being Element of the carrier of a1 holds
     b1,b2 // b2,b1) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     b1,b2 // b3,b3) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1 <> b2 & b1,b2 // b3,b4 & b1,b2 // b5,b6
        holds b3,b4 // b5,b6) &
     (for b1, b2, b3 being Element of the carrier of a1
           st b1,b2 // b1,b3
        holds b2,b1 // b2,b3) &
     (ex b1, b2, b3 being Element of the carrier of a1 st
        not b1,b2 // b1,b3) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b2 // b3,b4 & b1,b3 // b2,b4) &
     (for b1, b2 being Element of the carrier of a1 holds
     ex b3 being Element of the carrier of a1 st
        for b4, b5 being Element of the carrier of a1 holds
        b1,b2 // b1,b3 &
         (ex b6 being Element of the carrier of a1 st
            (b1,b3 // b1,b4 implies b1,b5 // b1,b6 & b3,b5 // b4,b6))) &
     (for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
           st not b1,b2 // b1,b4 & not b1,b2 // b1,b6 & b1,b2 // b1,b3 & b1,b4 // b1,b5 & b1,b6 // b1,b7 & b2,b4 // b3,b5 & b2,b6 // b3,b7
        holds b4,b6 // b5,b7) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st not b1,b2 // b1,b3 & not b1,b2 // b1,b5 & b1,b2 // b3,b4 & b1,b2 // b5,b6 & b1,b3 // b2,b4 & b1,b5 // b2,b6
        holds b3,b5 // b4,b6) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b1,b3 & b4,b5 // b4,b6 & b1,b5 // b2,b4 & b2,b6 // b3,b5
        holds b3,b4 // b1,b6) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st not b1,b2 // b1,b3 & b1,b2 // b3,b4 & b1,b3 // b2,b4
        holds not b1,b4 // b2,b3);
end;

:: SEMI_AF1:dfs 1
definiens
  let a1 be non empty AffinStruct;
To prove
     a1 is Semi_Affine_Space-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, b3 being Element of the carrier of a1 holds
     b1,b2 // b3,b3) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1 <> b2 & b1,b2 // b3,b4 & b1,b2 // b5,b6
        holds b3,b4 // b5,b6) &
     (for b1, b2, b3 being Element of the carrier of a1
           st b1,b2 // b1,b3
        holds b2,b1 // b2,b3) &
     (ex b1, b2, b3 being Element of the carrier of a1 st
        not b1,b2 // b1,b3) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b2 // b3,b4 & b1,b3 // b2,b4) &
     (for b1, b2 being Element of the carrier of a1 holds
     ex b3 being Element of the carrier of a1 st
        for b4, b5 being Element of the carrier of a1 holds
        b1,b2 // b1,b3 &
         (ex b6 being Element of the carrier of a1 st
            (b1,b3 // b1,b4 implies b1,b5 // b1,b6 & b3,b5 // b4,b6))) &
     (for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
           st not b1,b2 // b1,b4 & not b1,b2 // b1,b6 & b1,b2 // b1,b3 & b1,b4 // b1,b5 & b1,b6 // b1,b7 & b2,b4 // b3,b5 & b2,b6 // b3,b7
        holds b4,b6 // b5,b7) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st not b1,b2 // b1,b3 & not b1,b2 // b1,b5 & b1,b2 // b3,b4 & b1,b2 // b5,b6 & b1,b3 // b2,b4 & b1,b5 // b2,b6
        holds b3,b5 // b4,b6) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b1,b3 & b4,b5 // b4,b6 & b1,b5 // b2,b4 & b2,b6 // b3,b5
        holds b3,b4 // b1,b6) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st not b1,b2 // b1,b3 & b1,b2 // b3,b4 & b1,b3 // b2,b4
        holds not b1,b4 // b2,b3);

:: SEMI_AF1:def 1
theorem
for b1 being non empty AffinStruct holds
      b1 is Semi_Affine_Space-like
   iff
      (for b2, b3 being Element of the carrier of b1 holds
       b2,b3 // b3,b2) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       b2,b3 // b4,b4) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
             st b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7
          holds b4,b5 // b6,b7) &
       (for b2, b3, b4 being Element of the carrier of b1
             st b2,b3 // b2,b4
          holds b3,b2 // b3,b4) &
       (ex b2, b3, b4 being Element of the carrier of b1 st
          not b2,b3 // b2,b4) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b2,b3 // b4,b5 & b2,b4 // b3,b5) &
       (for b2, b3 being Element of the carrier of b1 holds
       ex b4 being Element of the carrier of b1 st
          for b5, b6 being Element of the carrier of b1 holds
          b2,b3 // b2,b4 &
           (ex b7 being Element of the carrier of b1 st
              (b2,b4 // b2,b5 implies b2,b6 // b2,b7 & b4,b6 // b5,b7))) &
       (for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
             st not b2,b3 // b2,b5 & not b2,b3 // b2,b7 & b2,b3 // b2,b4 & b2,b5 // b2,b6 & b2,b7 // b2,b8 & b3,b5 // b4,b6 & b3,b7 // b4,b8
          holds b5,b7 // b6,b8) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
             st not b2,b3 // b2,b4 & not b2,b3 // b2,b6 & b2,b3 // b4,b5 & b2,b3 // b6,b7 & b2,b4 // b3,b5 & b2,b6 // b3,b7
          holds b4,b6 // b5,b7) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
             st b2,b3 // b2,b4 & b5,b6 // b5,b7 & b2,b6 // b3,b5 & b3,b7 // b4,b6
          holds b4,b5 // b2,b7) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st not b2,b3 // b2,b4 & b2,b3 // b4,b5 & b2,b4 // b3,b5
          holds not b2,b5 // b3,b4);

:: SEMI_AF1:exreg 1
registration
  cluster non empty Semi_Affine_Space-like AffinStruct;
end;

:: SEMI_AF1:modenot 1
definition
  mode Semi_Affine_Space is non empty Semi_Affine_Space-like AffinStruct;
end;

:: SEMI_AF1:th 12
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 // b2,b3;

:: SEMI_AF1:th 13
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b4,b5 // b2,b3;

:: SEMI_AF1:th 14
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b2 // b3,b4;

:: SEMI_AF1:th 15
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b3,b2 // b4,b5;

:: SEMI_AF1:th 16
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b2,b3 // b5,b4;

:: SEMI_AF1:th 17
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b3,b2 // b4,b5 & b2,b3 // b5,b4 & b3,b2 // b5,b4 & b4,b5 // b2,b3 & b5,b4 // b2,b3 & b4,b5 // b3,b2 & b5,b4 // b3,b2;

:: SEMI_AF1:th 18
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3 // b2,b4
   holds b2,b4 // b2,b3 & b3,b2 // b2,b4 & b2,b3 // b4,b2 & b2,b4 // b3,b2 & b3,b2 // b4,b2 & b4,b2 // b2,b3 & b4,b2 // b3,b2 & b3,b2 // b3,b4 & b2,b3 // b3,b4 & b3,b2 // b4,b3 & b3,b4 // b3,b2 & b2,b3 // b4,b3 & b4,b3 // b3,b2 & b3,b4 // b2,b3 & b4,b3 // b2,b3 & b4,b2 // b4,b3 & b2,b4 // b4,b3 & b4,b2 // b3,b4 & b2,b4 // b3,b4 & b4,b3 // b4,b2 & b3,b4 // b4,b2 & b4,b3 // b2,b4 & b3,b4 // b2,b4;

:: SEMI_AF1:th 20
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2 <> b3 & b4,b5 // b2,b3 & b2,b3 // b6,b7
   holds b4,b5 // b6,b7;

:: SEMI_AF1:th 21
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st not b2,b3 // b2,b4
   holds b2 <> b3 & b3 <> b4 & b4 <> b2;

:: SEMI_AF1:th 22
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3 // b4,b5
   holds b2 <> b3 & b4 <> b5;

:: SEMI_AF1:th 23
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b2,b4 & b3,b5 // b3,b4 & b5,b2 // b5,b4
   holds b2,b3 // b2,b5;

:: SEMI_AF1:th 25
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not b2,b3 // b2,b4 & b5 <> b6 & b5,b6 // b5,b2 & b5,b6 // b5,b3
   holds not b5,b6 // b5,b4;

:: SEMI_AF1:th 26
theorem
for b1 being non empty Semi_Affine_Space-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 b2,b3 // b2,b4;

:: SEMI_AF1:th 28
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st not b2,b3 // b2,b4
   holds not b2,b3 // b4,b2 & not b3,b2 // b2,b4 & not b3,b2 // b4,b2 & not b2,b4 // b2,b3 & not b2,b4 // b3,b2 & not b4,b2 // b2,b3 & not b4,b2 // b3,b2 & not b3,b2 // b3,b4 & not b3,b2 // b4,b3 & not b2,b3 // b3,b4 & not b2,b3 // b4,b3 & not b3,b4 // b3,b2 & not b3,b4 // b2,b3 & not b4,b3 // b2,b3 & not b4,b3 // b3,b2 & not b4,b3 // b4,b2 & not b4,b3 // b2,b4 & not b3,b4 // b4,b2 & not b3,b4 // b2,b4 & not b4,b2 // b4,b3 & not b4,b2 // b3,b4 & not b2,b4 // b3,b4 & not b2,b4 // b4,b3;

:: SEMI_AF1:th 29
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
      st not b2,b3 // b4,b5 & b2,b3 // b6,b7 & b4,b5 // b8,b9 & b6 <> b7 & b8 <> b9
   holds not b6,b7 // b8,b9;

:: SEMI_AF1:th 30
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not b2,b3 // b2,b4 & b2,b3 // b5,b6 & b2,b4 // b5,b7 & b3,b4 // b6,b7 & b5 <> b6
   holds not b5,b6 // b5,b7;

:: SEMI_AF1:th 31
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not b2,b3 // b2,b4 & b2,b4 // b5,b6 & b3,b4 // b5,b6
   holds b5 = b6;

:: SEMI_AF1:th 32
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3 // b2,b4 & b2,b4 // b2,b5 & b3,b4 // b3,b5
   holds b4 = b5;

:: SEMI_AF1:th 33
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st not b2,b3 // b2,b4 & b2,b3 // b5,b6 & b2,b4 // b5,b7 & b2,b4 // b5,b8 & b3,b4 // b6,b7 & b3,b4 // b6,b8
   holds b7 = b8;

:: SEMI_AF1:th 34
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st (b2 <> b3 & b4 <> b5 & (b2 = b4 implies b3 <> b5) implies b2 = b5 & b3 = b4)
   holds b2,b3 // b4,b5;

:: SEMI_AF1:th 35
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st (b2 <> b3 & b2 <> b4 implies b3 = b4)
   holds b2,b3 // b2,b4;

:: SEMI_AF1:prednot 1 => SEMI_AF1:pred 1
definition
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  pred A2,A3,A4 is_collinear means
    a2,a3 // a2,a4;
end;

:: SEMI_AF1:dfs 2
definiens
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
To prove
     a2,a3,a4 is_collinear
it is sufficient to prove
  thus a2,a3 // a2,a4;

:: SEMI_AF1:def 2
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   b2,b3,b4 is_collinear
iff
   b2,b3 // b2,b4;

:: SEMI_AF1:th 37
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3,b4 is_collinear
   holds b2,b4,b3 is_collinear & b3,b2,b4 is_collinear & b3,b4,b2 is_collinear & b4,b2,b3 is_collinear & b4,b3,b2 is_collinear;

:: SEMI_AF1:th 39
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3 // b5,b6 & b2,b4 // b5,b7 & b5 <> b6 & b5 <> b7
   holds not b5,b6,b7 is_collinear;

:: SEMI_AF1:th 40
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st (b2 <> b3 & b3 <> b4 implies b4 = b2)
   holds b2,b3,b4 is_collinear;

:: SEMI_AF1:th 41
theorem
for b1 being non empty Semi_Affine_Space-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 b2,b3,b4 is_collinear;

:: SEMI_AF1:th 42
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3,b4 is_collinear & b2,b3,b5 is_collinear
   holds b2,b3 // b4,b5;

:: SEMI_AF1:th 43
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3 // b4,b5
   holds not b2,b3,b5 is_collinear;

:: SEMI_AF1:th 44
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3 // b4,b5 & b4 <> b5 & b4,b5,b6 is_collinear
   holds not b2,b3,b6 is_collinear;

:: SEMI_AF1:th 45
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b4,b5 is_collinear
   holds b2 = b5;

:: SEMI_AF1:th 46
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2 <> b3 & b2 <> b4 & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b4,b6 is_collinear
   holds b3,b4 // b5,b6;

:: SEMI_AF1:th 48
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not b2,b3 // b4,b5 & b2,b3,b6 is_collinear & b2,b3,b7 is_collinear & b4,b5,b6 is_collinear & b4,b5,b7 is_collinear
   holds b6 = b7;

:: SEMI_AF1:th 49
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <> b3 & b2,b3,b4 is_collinear & b2,b3 // b4,b5
   holds b2,b4 // b3,b5;

:: SEMI_AF1:th 50
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <> b3 & b2,b3,b4 is_collinear & b2,b3 // b4,b5
   holds b4,b3 // b4,b5;

:: SEMI_AF1:th 51
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b4,b6 is_collinear & b2,b4,b7 is_collinear & b3,b4 // b5,b6 & b3,b4 // b5,b6 & b3,b4 // b5,b7
   holds b6 = b7;

:: SEMI_AF1:th 52
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <> b3 & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear
   holds b2,b4,b5 is_collinear;

:: SEMI_AF1:prednot 2 => SEMI_AF1:pred 2
definition
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred parallelogram A2,A3,A4,A5 means
    not a2,a3,a4 is_collinear & a2,a3 // a4,a5 & a2,a4 // a3,a5;
end;

:: SEMI_AF1:dfs 3
definiens
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     parallelogram a2,a3,a4,a5
it is sufficient to prove
  thus not a2,a3,a4 is_collinear & a2,a3 // a4,a5 & a2,a4 // a3,a5;

:: SEMI_AF1:def 3
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   parallelogram b2,b3,b4,b5
iff
   not b2,b3,b4 is_collinear & b2,b3 // b4,b5 & b2,b4 // b3,b5;

:: SEMI_AF1:th 54
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds b2 <> b3 & b2 <> b4 & b4 <> b3 & b2 <> b5 & b3 <> b5 & b4 <> b5;

:: SEMI_AF1:th 55
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds not b2,b3,b4 is_collinear & not b3,b2,b5 is_collinear & not b4,b5,b2 is_collinear & not b5,b4,b3 is_collinear;

:: SEMI_AF1:th 56
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds not b2,b3,b4 is_collinear & not b2,b4,b3 is_collinear & not b2,b3,b5 is_collinear & not b2,b5,b3 is_collinear & not b2,b4,b5 is_collinear & not b2,b5,b4 is_collinear & not b3,b2,b4 is_collinear & not b3,b4,b2 is_collinear & not b3,b2,b5 is_collinear & not b3,b5,b2 is_collinear & not b3,b4,b5 is_collinear & not b3,b5,b4 is_collinear & not b4,b2,b3 is_collinear & not b4,b3,b2 is_collinear & not b4,b2,b5 is_collinear & not b4,b5,b2 is_collinear & not b4,b3,b5 is_collinear & not b4,b5,b3 is_collinear & not b5,b2,b3 is_collinear & not b5,b3,b2 is_collinear & not b5,b2,b4 is_collinear & not b5,b4,b2 is_collinear & not b5,b3,b4 is_collinear & not b5,b4,b3 is_collinear;

:: SEMI_AF1:th 57
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5 & b2,b3,b6 is_collinear
   holds not b4,b5,b6 is_collinear;

:: SEMI_AF1:th 58
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds parallelogram b2,b4,b3,b5;

:: SEMI_AF1:th 59
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds parallelogram b4,b5,b2,b3;

:: SEMI_AF1:th 60
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds parallelogram b3,b2,b5,b4;

:: SEMI_AF1:th 61
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds parallelogram b2,b4,b3,b5 & parallelogram b4,b5,b2,b3 & parallelogram b3,b2,b5,b4 & parallelogram b4,b2,b5,b3 & parallelogram b5,b3,b4,b2 & parallelogram b3,b5,b2,b4;

:: SEMI_AF1:th 62
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear
   holds ex b5 being Element of the carrier of b1 st
      parallelogram b2,b3,b4,b5;

:: SEMI_AF1:th 63
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5 & parallelogram b2,b3,b4,b6
   holds b5 = b6;

:: SEMI_AF1:th 64
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds not b2,b5 // b3,b4;

:: SEMI_AF1:th 65
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds not parallelogram b2,b3,b5,b4;

:: SEMI_AF1:th 66
theorem
for b1 being non empty Semi_Affine_Space-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
      b2,b3,b4 is_collinear & b4 <> b2 & b4 <> b3;

:: SEMI_AF1:th 67
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5 & parallelogram b2,b3,b6,b7
   holds b4,b6 // b5,b7;

:: SEMI_AF1:th 68
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & parallelogram b5,b6,b2,b3 & parallelogram b5,b6,b4,b7
   holds parallelogram b2,b3,b4,b7;

:: SEMI_AF1:th 69
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2,b3,b4 is_collinear & b3 <> b4 & parallelogram b2,b5,b3,b6 & parallelogram b2,b5,b4,b7
   holds parallelogram b3,b6,b4,b7;

:: SEMI_AF1:th 70
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5 & parallelogram b2,b3,b6,b7 & parallelogram b4,b5,b8,b9
   holds b6,b8 // b7,b9;

:: SEMI_AF1:th 71
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st b2 <> b3
   holds ex b4, b5 being Element of the carrier of b1 st
      parallelogram b2,b4,b5,b3;

:: SEMI_AF1:prednot 3 => SEMI_AF1:pred 3
definition
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred congr A2,A3,A4,A5 means
    ((a2 = a3 implies a4 <> a5)) implies ex b1, b2 being Element of the carrier of a1 st
       parallelogram b1,b2,a2,a3 & parallelogram b1,b2,a4,a5;
end;

:: SEMI_AF1:dfs 4
definiens
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     congr a2,a3,a4,a5
it is sufficient to prove
  thus ((a2 = a3 implies a4 <> a5)) implies ex b1, b2 being Element of the carrier of a1 st
       parallelogram b1,b2,a2,a3 & parallelogram b1,b2,a4,a5;

:: SEMI_AF1:def 4
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   congr b2,b3,b4,b5
iff
   ((b2 = b3 implies b4 <> b5) implies ex b6, b7 being Element of the carrier of b1 st
      parallelogram b6,b7,b2,b3 & parallelogram b6,b7,b4,b5);

:: SEMI_AF1:th 73
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st congr b2,b2,b3,b4
   holds b3 = b4;

:: SEMI_AF1:th 74
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st congr b2,b3,b4,b4
   holds b2 = b3;

:: SEMI_AF1:th 75
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st congr b2,b3,b3,b2
   holds b2 = b3;

:: SEMI_AF1:th 76
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st congr b2,b3,b4,b5
   holds b2,b3 // b4,b5;

:: SEMI_AF1:th 77
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st congr b2,b3,b4,b5
   holds b2,b4 // b3,b5;

:: SEMI_AF1:th 78
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st congr b2,b3,b4,b5 & not b2,b3,b4 is_collinear
   holds parallelogram b2,b3,b4,b5;

:: SEMI_AF1:th 79
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds congr b2,b3,b4,b5;

:: SEMI_AF1:th 80
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st congr b2,b3,b4,b5 & b2,b3,b4 is_collinear & parallelogram b6,b7,b2,b3
   holds parallelogram b6,b7,b4,b5;

:: SEMI_AF1:th 81
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st congr b2,b3,b4,b5 & congr b2,b3,b4,b6
   holds b5 = b6;

:: SEMI_AF1:th 82
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
   congr b2,b3,b4,b5;

:: SEMI_AF1:th 84
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
congr b2,b3,b2,b3;

:: SEMI_AF1:th 85
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st congr b2,b3,b4,b5 & congr b2,b3,b6,b7
   holds congr b4,b5,b6,b7;

:: SEMI_AF1:th 86
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st congr b2,b3,b4,b5
   holds congr b4,b5,b2,b3;

:: SEMI_AF1:th 87
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st congr b2,b3,b4,b5
   holds congr b3,b2,b5,b4;

:: SEMI_AF1:th 88
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st congr b2,b3,b4,b5
   holds congr b2,b4,b3,b5;

:: SEMI_AF1:th 89
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st congr b2,b3,b4,b5
   holds congr b4,b5,b2,b3 & congr b3,b2,b5,b4 & congr b2,b4,b3,b5 & congr b5,b4,b3,b2 & congr b3,b5,b2,b4 & congr b4,b2,b5,b3 & congr b5,b3,b4,b2;

:: SEMI_AF1:th 90
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st congr b2,b3,b4,b5 & congr b3,b6,b5,b7
   holds congr b2,b6,b4,b7;

:: SEMI_AF1:th 91
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st congr b2,b3,b4,b5 & congr b6,b3,b4,b7
   holds congr b2,b6,b7,b5;

:: SEMI_AF1:th 92
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st congr b2,b3,b3,b4 & congr b5,b3,b3,b6
   holds congr b2,b5,b6,b4;

:: SEMI_AF1:th 93
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st congr b2,b3,b4,b5 & congr b6,b3,b4,b7
   holds b2,b6 // b5,b7;

:: SEMI_AF1:th 94
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st congr b2,b3,b3,b4 & congr b5,b3,b3,b6
   holds b2,b5 // b4,b6;

:: SEMI_AF1:funcnot 1 => SEMI_AF1:func 1
definition
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  func sum(A2,A3,A4) -> Element of the carrier of a1 means
    congr a4,a2,a3,it;
end;

:: SEMI_AF1:def 5
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b5 = sum(b2,b3,b4)
iff
   congr b4,b2,b3,b5;

:: SEMI_AF1:funcnot 2 => SEMI_AF1:func 2
definition
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  func opposite(A2,A3) -> Element of the carrier of a1 means
    sum(a2,it,a3) = a3;
end;

:: SEMI_AF1:def 6
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   b4 = opposite(b2,b3)
iff
   sum(b2,b4,b3) = b3;

:: SEMI_AF1:funcnot 3 => SEMI_AF1:func 3
definition
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  func diff(A2,A3,A4) -> Element of the carrier of a1 equals
    sum(a2,opposite(a3,a4),a4);
end;

:: SEMI_AF1:def 7
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
diff(b2,b3,b4) = sum(b2,opposite(b3,b4),b4);

:: SEMI_AF1:th 99
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
sum(b2,b3,b3) = b2;

:: SEMI_AF1:th 100
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
   sum(b2,b4,b3) = b3;

:: SEMI_AF1:th 101
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
sum(sum(b2,b3,b4),b5,b4) = sum(b2,sum(b3,b5,b4),b4);

:: SEMI_AF1:th 102
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
sum(b2,b3,b4) = sum(b3,b2,b4);

:: SEMI_AF1:th 103
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st sum(b2,b2,b3) = b3
   holds b2 = b3;

:: SEMI_AF1:th 104
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st sum(b2,b3,b4) = sum(b2,b5,b4)
   holds b3 = b5;

:: SEMI_AF1:th 106
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
congr b2,b3,b3,opposite(b2,b3);

:: SEMI_AF1:th 107
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st opposite(b2,b3) = opposite(b4,b3)
   holds b2 = b4;

:: SEMI_AF1:th 108
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 // opposite(b2,b4),opposite(b3,b4);

:: SEMI_AF1:th 109
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2 being Element of the carrier of b1 holds
   opposite(b2,b2) = b2;

:: SEMI_AF1:th 110
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2,b3 // sum(b2,b4,b5),sum(b3,b4,b5);

:: SEMI_AF1:th 111
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b2,b3 // sum(b2,b4,b6),sum(b3,b5,b6);

:: SEMI_AF1:th 113
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   diff(b2,b3,b4) = b4
iff
   b2 = b3;

:: SEMI_AF1:th 114
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,diff(b3,b4,b2) // b4,b3;

:: SEMI_AF1:th 115
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
   b2,diff(b3,b4,b2),diff(b5,b6,b2) is_collinear
iff
   b4,b3 // b6,b5;

:: SEMI_AF1:prednot 4 => SEMI_AF1:pred 4
definition
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4, a5, a6 be Element of the carrier of a1;
  pred trap A2,A3,A4,A5,A6 means
    not a6,a2,a4 is_collinear & a6,a2,a3 is_collinear & a6,a4,a5 is_collinear & a2,a4 // a3,a5;
end;

:: SEMI_AF1:dfs 8
definiens
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3, a4, a5, a6 be Element of the carrier of a1;
To prove
     trap a2,a3,a4,a5,a6
it is sufficient to prove
  thus not a6,a2,a4 is_collinear & a6,a2,a3 is_collinear & a6,a4,a5 is_collinear & a2,a4 // a3,a5;

:: SEMI_AF1:def 8
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
   trap b2,b3,b4,b5,b6
iff
   not b6,b2,b4 is_collinear & b6,b2,b3 is_collinear & b6,b4,b5 is_collinear & b2,b4 // b3,b5;

:: SEMI_AF1:prednot 5 => SEMI_AF1:pred 5
definition
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  pred qtrap A2,A3 means
    for b1, b2 being Element of the carrier of a1 holds
    ex b3 being Element of the carrier of a1 st
       (a2,a3,b1 is_collinear implies a2,b2,b3 is_collinear & a3,b2 // b1,b3);
end;

:: SEMI_AF1:dfs 9
definiens
  let a1 be non empty Semi_Affine_Space-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
To prove
     qtrap a2,a3
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1 holds
    ex b3 being Element of the carrier of a1 st
       (a2,a3,b1 is_collinear implies a2,b2,b3 is_collinear & a3,b2 // b1,b3);

:: SEMI_AF1:def 9
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
   qtrap b2,b3
iff
   for b4, b5 being Element of the carrier of b1 holds
   ex b6 being Element of the carrier of b1 st
      (b2,b3,b4 is_collinear implies b2,b5,b6 is_collinear & b3,b5 // b4,b6);

:: SEMI_AF1:th 118
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st trap b2,b3,b4,b5,b6
   holds b6 <> b2 & b2 <> b4 & b4 <> b6;

:: SEMI_AF1:th 119
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st trap b2,b3,b4,b5,b6 & trap b2,b3,b4,b7,b6
   holds b5 = b7;

:: SEMI_AF1:th 120
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear
   holds trap b3,b2,b4,b2,b2;

:: SEMI_AF1:th 121
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st trap b2,b3,b4,b5,b6
   holds trap b4,b5,b2,b3,b6;

:: SEMI_AF1:th 122
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st trap b2,b3,b4,b5,b5
   holds b5 = b3;

:: SEMI_AF1:th 123
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2 <> b3 & trap b4,b3,b5,b6,b2
   holds not b2,b3,b6 is_collinear;

:: SEMI_AF1:th 124
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2 <> b3 & trap b4,b3,b5,b6,b2
   holds trap b3,b4,b6,b5,b2;

:: SEMI_AF1:th 125
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st trap b2,b3,b4,b5,b3
   holds b3 = b5;

:: SEMI_AF1:th 126
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st trap b2,b3,b4,b5,b6 & trap b2,b3,b7,b8,b6
   holds b4,b7 // b5,b8;

:: SEMI_AF1:th 127
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st trap b2,b3,b4,b5,b6 & trap b2,b3,b7,b8,b6 & not b6,b4,b7 is_collinear
   holds trap b4,b5,b7,b8,b6;

:: SEMI_AF1:th 128
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
      st trap b2,b3,b4,b5,b6 & trap b2,b3,b7,b8,b6 & trap b4,b5,b9,b10,b6
   holds b7,b9 // b8,b10;

:: SEMI_AF1:th 129
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
   b2,b3,b4 is_collinear & qtrap b2,b4;

:: SEMI_AF1:th 130
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct holds
   ex b2, b3, b4 being Element of the carrier of b1 st
      b2 <> b3 & b3 <> b4 & b4 <> b2;

:: SEMI_AF1:th 131
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st qtrap b2,b3
   holds b2 <> b3;

:: SEMI_AF1:th 132
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st qtrap b2,b3
   holds ex b4 being Element of the carrier of b1 st
      not b2,b3,b4 is_collinear & qtrap b2,b4;

:: SEMI_AF1:th 133
theorem
for b1 being non empty Semi_Affine_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & qtrap b2,b3
   holds ex b6 being Element of the carrier of b1 st
      trap b3,b5,b4,b6,b2;