Article PARSP_2, MML version 4.99.1005

:: PARSP_2:th 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   MPS b1 is non empty ParSp-like ParStr;

:: PARSP_2:th 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of MPS b1 holds
   b2,b3 '||' b4,b5
iff
   ex b6, b7, b8, b9 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] st
      [b2,b3,b4,b5] = [b6,b7,b8,b9] &
       (for b10 being Element of the carrier of b1
             st b10 * (b6 `1 - (b7 `1)) = b8 `1 - (b9 `1) &
                b10 * (b6 `2 - (b7 `2)) = b8 `2 - (b9 `2)
          holds b10 * (b6 `3 - (b7 `3)) <> b8 `3 - (b9 `3) implies b6 `1 - (b7 `1) = 0. b1 & b6 `2 - (b7 `2) = 0. b1 & b6 `3 - (b7 `3) = 0. b1);

:: PARSP_2:th 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4 being Element of the carrier of MPS b1
for b5, b6, b7 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:]
      st not b2,b3 '||' b2,b4 &
         [b2,b3,b2,b4] = [b5,b6,b5,b7]
   holds b5 <> b6 & b5 <> b7 & b6 <> b7;

:: PARSP_2:th 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4 being Element of the carrier of MPS b1
for b5, b6, b7 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:]
for b8, b9 being Element of the carrier of b1
      st not b2,b3 '||' b2,b4 &
         [b2,b3,b2,b4] = [b5,b6,b5,b7] &
         b8 * (b5 `1 - (b6 `1)) = b9 * (b5 `1 - (b7 `1)) &
         b8 * (b5 `2 - (b6 `2)) = b9 * (b5 `2 - (b7 `2)) &
         b8 * (b5 `3 - (b6 `3)) = b9 * (b5 `3 - (b7 `3))
   holds b8 = 0. b1 & b9 = 0. b1;

:: PARSP_2:th 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of MPS b1
for b6, b7, b8, b9 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:]
      st not b2,b3 '||' b2,b4 &
         b2,b3 '||' b4,b5 &
         b2,b4 '||' b3,b5 &
         [b2,b3,b4,b5] = [b6,b7,b8,b9]
   holds b9 `1 = (b7 `1 + (b8 `1)) - (b6 `1) &
    b9 `2 = (b7 `2 + (b8 `2)) - (b6 `2) &
    b9 `3 = (b7 `3 + (b8 `3)) - (b6 `3);

:: PARSP_2:th 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   ex b2, b3, b4 being Element of the carrier of MPS b1 st
      not b2,b3 '||' b2,b4;

:: PARSP_2:th 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of MPS b1
      st (1_ b1) + 1_ b1 <> 0. b1 & b2,b3 '||' b4,b5 & b4,b2 '||' b3,b5 & b4,b3 '||' b2,b5
   holds b4,b2 '||' b4,b3;

:: PARSP_2:th 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of MPS b1
      st not b2,b3 '||' b2,b4 & not b2,b3 '||' b2,b5 & b2,b3 '||' b4,b6 & b2,b3 '||' b5,b7 & b2,b4 '||' b3,b6 & b2,b5 '||' b3,b7
   holds b4,b5 '||' b6,b7;

:: PARSP_2:attrnot 1 => PARSP_2:attr 1
definition
  let a1 be non empty ParSp-like ParStr;
  attr a1 is FanodesSp-like means
    (ex b1, b2, b3 being Element of the carrier of a1 st
        not b1,b2 '||' b1,b3) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b2,b3 '||' b1,b4 & b1,b2 '||' b3,b4 & b1,b3 '||' b2,b4
        holds b1,b2 '||' b1,b3) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st not b1,b4 '||' b1,b2 & not b1,b4 '||' b1,b3 & b1,b4 '||' b2,b5 & b1,b4 '||' b3,b6 & b1,b2 '||' b4,b5 & b1,b3 '||' b4,b6
        holds b2,b3 '||' b5,b6);
end;

:: PARSP_2:dfs 1
definiens
  let a1 be non empty ParSp-like ParStr;
To prove
     a1 is FanodesSp-like
it is sufficient to prove
  thus (ex b1, b2, b3 being Element of the carrier of a1 st
        not b1,b2 '||' b1,b3) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b2,b3 '||' b1,b4 & b1,b2 '||' b3,b4 & b1,b3 '||' b2,b4
        holds b1,b2 '||' b1,b3) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st not b1,b4 '||' b1,b2 & not b1,b4 '||' b1,b3 & b1,b4 '||' b2,b5 & b1,b4 '||' b3,b6 & b1,b2 '||' b4,b5 & b1,b3 '||' b4,b6
        holds b2,b3 '||' b5,b6);

:: PARSP_2:def 1
theorem
for b1 being non empty ParSp-like ParStr holds
      b1 is FanodesSp-like
   iff
      (ex b2, b3, b4 being Element of the carrier of b1 st
          not b2,b3 '||' b2,b4) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b3,b4 '||' b2,b5 & b2,b3 '||' b4,b5 & b2,b4 '||' b3,b5
          holds b2,b3 '||' b2,b4) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
             st not b2,b5 '||' b2,b3 & not b2,b5 '||' b2,b4 & b2,b5 '||' b3,b6 & b2,b5 '||' b4,b7 & b2,b3 '||' b5,b6 & b2,b4 '||' b5,b7
          holds b3,b4 '||' b6,b7);

:: PARSP_2:exreg 1
registration
  cluster non empty strict ParSp-like FanodesSp-like ParStr;
end;

:: PARSP_2:modenot 1
definition
  mode FanodesSp is non empty ParSp-like FanodesSp-like ParStr;
end;

:: PARSP_2:th 13
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:prednot 1 => PARSP_2:pred 1
definition
  let a1 be non empty ParSp-like FanodesSp-like ParStr;
  let a2, a3, a4 be Element of the carrier of a1;
  pred A2,A3,A4 is_collinear means
    a2,a3 '||' a2,a4;
end;

:: PARSP_2:dfs 2
definiens
  let a1 be non empty ParSp-like FanodesSp-like ParStr;
  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;

:: PARSP_2:def 2
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4 being Element of the carrier of b1 holds
   b2,b3,b4 is_collinear
iff
   b2,b3 '||' b2,b4;

:: PARSP_2:th 15
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3,b4 is_collinear
   holds b2,b4,b3 is_collinear & b4,b3,b2 is_collinear & b3,b2,b4 is_collinear & b3,b4,b2 is_collinear & b4,b2,b3 is_collinear;

:: PARSP_2:th 17
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 18
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 19
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2 <> b3 & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b3,b6 is_collinear
   holds b4,b5,b6 is_collinear;

:: PARSP_2:th 20
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 21
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 22
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 23
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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 & b2,b3,b6 is_collinear
   holds not b4,b5,b6 is_collinear;

:: PARSP_2:th 24
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 25
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 26
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 27
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 28
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 29
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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,b7
   holds b6 = b7;

:: PARSP_2:th 30
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 31
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3,b4 is_collinear & b2,b4,b5 is_collinear & b2 <> b4
   holds b3,b4,b5 is_collinear;

:: PARSP_2:prednot 2 => PARSP_2:pred 2
definition
  let a1 be non empty ParSp-like FanodesSp-like ParStr;
  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;

:: PARSP_2:dfs 3
definiens
  let a1 be non empty ParSp-like FanodesSp-like ParStr;
  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;

:: PARSP_2:def 3
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 34
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds b2 <> b3 & b3 <> b4 & b4 <> b2 & b2 <> b5 & b3 <> b5 & b4 <> b5;

:: PARSP_2:th 35
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 36
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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 & not b2,b4,b3 is_collinear & not b3,b2,b4 is_collinear & not b3,b4,b2 is_collinear & not b4,b2,b3 is_collinear & not b4,b3,b2 is_collinear & not b3,b5,b2 is_collinear & not b2,b3,b5 is_collinear & not b2,b5,b3 is_collinear & not b5,b2,b3 is_collinear & not b5,b3,b2 is_collinear & not b4,b2,b5 is_collinear & not b2,b4,b5 is_collinear & not b2,b5,b4 is_collinear & not b5,b2,b4 is_collinear & not b5,b4,b2 is_collinear & not b5,b3,b4 is_collinear & not b3,b4,b5 is_collinear & not b3,b5,b4 is_collinear & not b4,b3,b5 is_collinear & not b4,b5,b3 is_collinear;

:: PARSP_2:th 37
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 38
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds parallelogram b2,b4,b3,b5;

:: PARSP_2:th 39
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds parallelogram b4,b5,b2,b3;

:: PARSP_2:th 40
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds parallelogram b3,b2,b5,b4;

:: PARSP_2:th 41
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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 & parallelogram b5,b4,b3,b2;

:: PARSP_2:th 42
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 43
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 44
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds not b2,b5 '||' b3,b4;

:: PARSP_2:th 45
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 46
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 47
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 48
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 49
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 50
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:th 51
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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,b3,b4,b5;

:: PARSP_2:th 52
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
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;

:: PARSP_2:prednot 3 => PARSP_2:pred 3
definition
  let a1 be non empty ParSp-like FanodesSp-like ParStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3 congr 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;

:: PARSP_2:dfs 4
definiens
  let a1 be non empty ParSp-like FanodesSp-like ParStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3 congr 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;

:: PARSP_2:def 4
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3 congr 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);

:: PARSP_2:th 55
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4 being Element of the carrier of b1
      st b2,b2 congr b3,b4
   holds b3 = b4;

:: PARSP_2:th 56
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3 congr b4,b4
   holds b2 = b3;

:: PARSP_2:th 57
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3 being Element of the carrier of b1
      st b2,b3 congr b3,b2
   holds b2 = b3;

:: PARSP_2:th 58
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 congr b4,b5
   holds b2,b3 '||' b4,b5;

:: PARSP_2:th 59
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 congr b4,b5
   holds b2,b4 '||' b3,b5;

:: PARSP_2:th 60
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 congr b4,b5 & not b2,b3,b4 is_collinear
   holds parallelogram b2,b3,b4,b5;

:: PARSP_2:th 61
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st parallelogram b2,b3,b4,b5
   holds b2,b3 congr b4,b5;

:: PARSP_2:th 62
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2,b3 congr b4,b5 & b2,b3,b4 is_collinear & parallelogram b6,b7,b2,b3
   holds parallelogram b6,b7,b4,b5;

:: PARSP_2:th 63
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2,b3 congr b4,b5 & b2,b3 congr b4,b6
   holds b5 = b6;

:: PARSP_2:th 64
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
   b2,b3 congr b4,b5;

:: PARSP_2:th 66
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3 being Element of the carrier of b1 holds
b2,b3 congr b2,b3;

:: PARSP_2:th 67
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2,b3 congr b4,b5 & b2,b3 congr b6,b7
   holds b4,b5 congr b6,b7;

:: PARSP_2:th 68
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 congr b4,b5
   holds b4,b5 congr b2,b3;

:: PARSP_2:th 69
theorem
for b1 being non empty ParSp-like FanodesSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 congr b4,b5
   holds b3,b2 congr b5,b4;