:: HOMOTHET semantic presentation

Lemma1: for b1 being AffinPlane holds
( b1 satisfies_DES iff for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
( LIN b2,b3,b4 & LIN b2,b5,b6 & LIN b2,b7,b8 & not LIN b2,b3,b5 & not LIN b2,b3,b7 & b3,b5 // b4,b6 & b3,b7 // b4,b8 implies b5,b7 // b6,b8 ) )
proof end;

Lemma2: for b1 being AffinPlane holds
( b1 satisfies_TDES iff for b2 being Element of b1
for b3 being Subset of b1
for b4, b5, b6, b7, b8, b9 being Element of b1 holds
( b2 in b3 & b4 in b3 & b5 in b3 & b3 is_line & LIN b2,b6,b8 & LIN b2,b7,b9 & not b6 in b3 & b6,b7 // b3 & b8,b9 // b3 & b6,b4 // b8,b5 implies b7,b4 // b9,b5 ) )
proof end;

Lemma3: for b1 being AffinPlane holds
( b1 satisfies_TDES implies for b2 being Subset of b1
for b3, b4, b5, b6, b7, b8, b9 being Element of b1 holds
( b2 is_line & b3 in b2 & b6 in b2 & b9 in b2 & not b4 in b2 & b4 <> b5 & LIN b3,b4,b7 & b4,b5 // b7,b8 & b5,b6 // b8,b9 & b4,b6 // b7,b9 & b4,b5 // b2 implies LIN b3,b5,b8 ) )
proof end;

Lemma4: for b1 being AffinPlane holds
( b1 satisfies_TDES implies for b2, b3, b4 being Subset of b1
for b5, b6, b7, b8, b9, b10 being Element of b1 holds
( b2 // b3 & b2 // b4 & b2 <> b3 & b2 <> b4 & b5 in b2 & b6 in b2 & b7 in b3 & b8 in b3 & b9 in b4 & b10 in b4 & b5,b7 // b6,b8 & b5,b9 // b6,b10 implies b7,b9 // b8,b10 ) )
proof end;

theorem Th1: :: HOMOTHET:1
for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of b1 holds
( not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b3,b6 & LIN b2,b3,b7 & LIN b2,b4,b8 & LIN b2,b4,b9 & LIN b2,b4,b10 & b4 <> b9 & b3 <> b6 & b2 <> b9 & b2 <> b6 & b3,b4 // b5,b8 & b3,b9 // b5,b10 & b6,b4 // b7,b8 & b1 satisfies_DES implies b6,b9 // b7,b10 )
proof end;

theorem Th2: :: HOMOTHET:2
for b1 being AffinPlane holds
( ( for b2, b3, b4 being Element of b1 holds
not ( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & ( for b5 being Permutation of the carrier of b1 holds
not ( b5 is_Dil & b5 . b2 = b2 & b5 . b3 = b4 ) ) ) ) implies b1 satisfies_DES )
proof end;

Lemma7: for b1 being AffinPlane
for b2, b3, b4, b5, b6 being Element of b1 holds
( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & LIN b2,b3,b5 & ( ( not LIN b2,b3,b6 & LIN b2,b6,b5 & b3,b6 // b4,b5 ) or ( LIN b2,b3,b6 & ex b7, b8 being Element of b1 st
( not LIN b2,b3,b7 & LIN b2,b7,b8 & b3,b7 // b4,b8 & b7,b6 // b8,b5 & LIN b2,b3,b5 ) ) ) implies LIN b2,b3,b6 )
proof end;

Lemma8: for b1 being AffinPlane
for b2, b3, b4, b5, b6 being Element of b1 holds
( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & ( ( not LIN b2,b3,b5 & LIN b2,b5,b6 & b3,b5 // b4,b6 ) or ( LIN b2,b3,b5 & ex b7, b8 being Element of b1 st
( not LIN b2,b3,b7 & LIN b2,b7,b8 & b3,b7 // b4,b8 & b7,b5 // b8,b6 & LIN b2,b3,b6 ) ) ) implies ( b2 <> b4 & b2 <> b3 & LIN b2,b4,b3 & ( ( not LIN b2,b4,b6 & LIN b2,b6,b5 & b4,b6 // b3,b5 ) or ( LIN b2,b4,b6 & ex b7, b8 being Element of b1 st
( not LIN b2,b4,b7 & LIN b2,b7,b8 & b4,b7 // b3,b8 & b7,b6 // b8,b5 & LIN b2,b4,b5 ) ) ) ) )
proof end;

Lemma9: for b1 being AffinPlane
for b2, b3, b4, b5, b6 being Element of b1 holds
( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & b5 = b2 & ( ( not LIN b2,b3,b5 & LIN b2,b5,b6 & b3,b5 // b4,b6 ) or ( LIN b2,b3,b5 & ex b7, b8 being Element of b1 st
( not LIN b2,b3,b7 & LIN b2,b7,b8 & b3,b7 // b4,b8 & b7,b5 // b8,b6 & LIN b2,b3,b6 ) ) ) implies b6 = b2 )
proof end;

Lemma10: for b1 being AffinPlane
for b2, b3, b4, b5 being Element of b1 holds
not ( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & ( for b6 being Element of b1 holds
( not ( not LIN b2,b3,b5 & LIN b2,b5,b6 & b3,b5 // b4,b6 ) & not ( LIN b2,b3,b5 & ex b7, b8 being Element of b1 st
( not LIN b2,b3,b7 & LIN b2,b7,b8 & b3,b7 // b4,b8 & b7,b5 // b8,b6 & LIN b2,b3,b6 ) ) ) ) )
proof end;

Lemma11: for b1 being AffinPlane
for b2, b3, b4, b5 being Element of b1 holds
not ( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & ( for b6 being Element of b1 holds
( not ( not LIN b2,b3,b6 & LIN b2,b6,b5 & b3,b6 // b4,b5 ) & not ( LIN b2,b3,b6 & ex b7, b8 being Element of b1 st
( not LIN b2,b3,b7 & LIN b2,b7,b8 & b3,b7 // b4,b8 & b7,b6 // b8,b5 & LIN b2,b3,b5 ) ) ) ) )
proof end;

Lemma12: for b1 being AffinPlane
for b2, b3, b4, b5, b6 being Element of b1 holds
( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & b3 = b4 & ( ( not LIN b2,b3,b5 & LIN b2,b5,b6 & b3,b5 // b4,b6 ) or ( LIN b2,b3,b5 & ex b7, b8 being Element of b1 st
( not LIN b2,b3,b7 & LIN b2,b7,b8 & b3,b7 // b4,b8 & b7,b5 // b8,b6 & LIN b2,b3,b6 ) ) ) implies b5 = b6 )
proof end;

Lemma13: for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & b1 satisfies_DES & LIN b2,b3,b5 & ex b8, b9 being Element of b1 st
( not LIN b2,b3,b8 & LIN b2,b8,b9 & b3,b8 // b4,b9 & b8,b5 // b9,b6 & LIN b2,b3,b6 ) & ex b8, b9 being Element of b1 st
( not LIN b2,b3,b8 & LIN b2,b8,b9 & b3,b8 // b4,b9 & b8,b5 // b9,b7 & LIN b2,b3,b7 ) implies b6 = b7 )
proof end;

Lemma14: for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & b1 satisfies_DES & ( ( not LIN b2,b3,b5 & LIN b2,b5,b6 & b3,b5 // b4,b6 ) or ( LIN b2,b3,b5 & ex b8, b9 being Element of b1 st
( not LIN b2,b3,b8 & LIN b2,b8,b9 & b3,b8 // b4,b9 & b8,b5 // b9,b6 & LIN b2,b3,b6 ) ) ) & ( ( not LIN b2,b3,b7 & LIN b2,b7,b6 & b3,b7 // b4,b6 ) or ( LIN b2,b3,b7 & ex b8, b9 being Element of b1 st
( not LIN b2,b3,b8 & LIN b2,b8,b9 & b3,b8 // b4,b9 & b8,b7 // b9,b6 & LIN b2,b3,b6 ) ) ) implies b5 = b7 )
proof end;

Lemma15: for b1 being AffinPlane
for b2, b3, b4 being Element of b1 holds
not ( b1 satisfies_DES & b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & ( for b5 being Permutation of the carrier of b1 holds
not for b6, b7 being Element of b1 holds
( b5 . b6 = b7 iff ( ( not LIN b2,b3,b6 & LIN b2,b6,b7 & b3,b6 // b4,b7 ) or ( LIN b2,b3,b6 & ex b8, b9 being Element of b1 st
( not LIN b2,b3,b8 & LIN b2,b8,b9 & b3,b8 // b4,b9 & b8,b6 // b9,b7 & LIN b2,b3,b7 ) ) ) ) ) )
proof end;

Lemma16: for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
( b1 satisfies_DES & b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & not LIN b2,b3,b5 & LIN b2,b5,b6 & b3,b5 // b4,b6 & LIN b2,b3,b7 & ex b9, b10 being Element of b1 st
( not LIN b2,b3,b9 & LIN b2,b9,b10 & b3,b9 // b4,b10 & b9,b7 // b10,b8 & LIN b2,b3,b8 ) implies b5,b7 // b6,b8 )
proof end;

Lemma17: for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
( b1 satisfies_DES & b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & LIN b2,b3,b5 & ex b9, b10 being Element of b1 st
( not LIN b2,b3,b9 & LIN b2,b9,b10 & b3,b9 // b4,b10 & b9,b5 // b10,b6 & LIN b2,b3,b6 ) & not LIN b2,b3,b7 & LIN b2,b7,b8 & b3,b7 // b4,b8 implies b5,b7 // b6,b8 )
proof end;

Lemma18: for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & LIN b2,b3,b5 & ex b9, b10 being Element of b1 st
( not LIN b2,b3,b9 & LIN b2,b9,b10 & b3,b9 // b4,b10 & b9,b5 // b10,b6 & LIN b2,b3,b6 ) & LIN b2,b3,b7 & ex b9, b10 being Element of b1 st
( not LIN b2,b3,b9 & LIN b2,b9,b10 & b3,b9 // b4,b10 & b9,b7 // b10,b8 & LIN b2,b3,b8 ) implies b5,b7 // b6,b8 )
proof end;

Lemma19: for b1 being AffinPlane
for b2, b3, b4 being Element of b1
for b5 being Permutation of the carrier of b1 holds
( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & b1 satisfies_DES & ( for b6, b7 being Element of b1 holds
( b5 . b6 = b7 iff ( ( not LIN b2,b3,b6 & LIN b2,b6,b7 & b3,b6 // b4,b7 ) or ( LIN b2,b3,b6 & ex b8, b9 being Element of b1 st
( not LIN b2,b3,b8 & LIN b2,b8,b9 & b3,b8 // b4,b9 & b8,b6 // b9,b7 & LIN b2,b3,b7 ) ) ) ) ) implies ( b5 is_Dil & b5 . b2 = b2 & b5 . b3 = b4 ) )
proof end;

theorem Th3: :: HOMOTHET:3
for b1 being AffinPlane holds
( b1 satisfies_DES implies for b2, b3, b4 being Element of b1 holds
not ( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & ( for b5 being Permutation of the carrier of b1 holds
not ( b5 is_Dil & b5 . b2 = b2 & b5 . b3 = b4 ) ) ) )
proof end;

theorem Th4: :: HOMOTHET:4
for b1 being AffinPlane holds
( b1 satisfies_DES iff for b2, b3, b4 being Element of b1 holds
not ( b2 <> b3 & b2 <> b4 & LIN b2,b3,b4 & ( for b5 being Permutation of the carrier of b1 holds
not ( b5 is_Dil & b5 . b2 = b2 & b5 . b3 = b4 ) ) ) ) by Th2, Th3;

definition
let c1 be AffinPlane;
let c2 be Permutation of the carrier of c1;
let c3 be Subset of c1;
pred c2 is_Sc c3 means :Def1: :: HOMOTHET:def 1
( a2 is_Col & a3 is_line & ( for b1 being Element of a1 holds
( b1 in a3 implies a2 . b1 = b1 ) ) & ( for b1 being Element of a1 holds b1,a2 . b1 // a3 ) );
end;

:: deftheorem Def1 defines is_Sc HOMOTHET:def 1 :
for b1 being AffinPlane
for b2 being Permutation of the carrier of b1
for b3 being Subset of b1 holds
( b2 is_Sc b3 iff ( b2 is_Col & b3 is_line & ( for b4 being Element of b1 holds
( b4 in b3 implies b2 . b4 = b4 ) ) & ( for b4 being Element of b1 holds b4,b2 . b4 // b3 ) ) );

theorem Th5: :: HOMOTHET:5
for b1 being AffinPlane
for b2 being Element of b1
for b3 being Subset of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Sc b3 & b4 . b2 = b2 & not b2 in b3 implies b4 = id the carrier of b1 )
proof end;

theorem Th6: :: HOMOTHET:6
for b1 being AffinPlane holds
( ( for b2, b3 being Element of b1
for b4 being Subset of b1 holds
not ( b2,b3 // b4 & not b2 in b4 & ( for b5 being Permutation of the carrier of b1 holds
not ( b5 is_Sc b4 & b5 . b2 = b3 ) ) ) ) implies b1 satisfies_TDES )
proof end;

Lemma24: for b1 being AffinPlane
for b2, b3, b4, b5 being Element of b1
for b6 being Subset of b1 holds
( b2,b3 // b6 & not b2 in b6 & ( ( b4 in b6 & b4 = b5 ) or ( not b4 in b6 & ex b7, b8 being Element of b1 st
( b7 in b6 & b8 in b6 & b7,b2 // b8,b4 & b7,b3 // b8,b5 & b4,b5 // b6 ) ) ) implies ( b3,b2 // b6 & not b3 in b6 & ( ( b5 in b6 & b5 = b4 ) or ( not b5 in b6 & ex b7, b8 being Element of b1 st
( b7 in b6 & b8 in b6 & b7,b3 // b8,b5 & b7,b2 // b8,b4 & b5,b4 // b6 ) ) ) ) )
proof end;

Lemma25: for b1 being AffinPlane
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b2,b3 // b4 & not b2 in b4 implies for b5 being Element of b1 holds
not for b6 being Element of b1 holds
( not ( b5 in b4 & b5 = b6 ) & not ( not b5 in b4 & ex b7, b8 being Element of b1 st
( b7 in b4 & b8 in b4 & b7,b2 // b8,b5 & b7,b3 // b8,b6 & b5,b6 // b4 ) ) ) )
proof end;

Lemma26: for b1 being AffinPlane
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
( b2,b3 // b4 & not b2 in b4 implies for b5 being Element of b1 holds
not for b6 being Element of b1 holds
( not ( b6 in b4 & b6 = b5 ) & not ( not b6 in b4 & ex b7, b8 being Element of b1 st
( b7 in b4 & b8 in b4 & b7,b2 // b8,b6 & b7,b3 // b8,b5 & b6,b5 // b4 ) ) ) )
proof end;

theorem Th7: :: HOMOTHET:7
for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of b1
for b10, b11 being Subset of b1 holds
( b10 // b11 & b2 in b10 & b3 in b10 & b4 in b10 & b5 in b10 & b1 satisfies_TDES & b6 in b11 & b7 in b11 & b8 in b11 & b9 in b11 & b6 <> b7 & b3 <> b2 & b2,b6 // b4,b8 & b2,b7 // b4,b9 & b3,b6 // b5,b8 implies b3,b7 // b5,b9 )
proof end;

Lemma28: for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of b1
for b10 being Subset of b1 holds
( b2,b3 // b10 & not b2 in b10 & not b4 in b10 & b1 satisfies_TDES & b5 in b10 & b6 in b10 & b5,b2 // b6,b4 & b5,b3 // b6,b7 & b4,b7 // b10 & b8 in b10 & b9 in b10 & b8,b2 // b9,b4 implies b8,b3 // b9,b7 )
proof end;

Lemma29: for b1 being AffinPlane
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of b1
for b11 being Subset of b1 holds
( b2,b3 // b11 & not b2 in b11 & not b4 in b11 & b1 satisfies_TDES & b5 in b11 & b6 in b11 & b5,b2 // b6,b4 & b5,b3 // b6,b7 & b4,b7 // b11 & b8 in b11 & b9 in b11 & b8,b2 // b9,b4 & b4,b10 // b11 & b8,b3 // b9,b10 implies b10 = b7 )
proof end;

Lemma30: for b1 being AffinPlane
for b2, b3, b4, b5, b6 being Element of b1
for b7 being Subset of b1 holds
( b2,b3 // b7 & not b2 in b7 & b1 satisfies_TDES & ( ( b4 in b7 & b4 = b5 ) or ( not b4 in b7 & ex b8, b9 being Element of b1 st
( b8 in b7 & b9 in b7 & b8,b2 // b9,b4 & b8,b3 // b9,b5 & b4,b5 // b7 ) ) ) & ( ( b6 in b7 & b6 = b5 ) or ( not b6 in b7 & ex b8, b9 being Element of b1 st
( b8 in b7 & b9 in b7 & b8,b2 // b9,b6 & b8,b3 // b9,b5 & b6,b5 // b7 ) ) ) implies b4 = b6 )
proof end;

Lemma31: for b1 being AffinPlane
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
not ( b2,b3 // b4 & not b2 in b4 & b1 satisfies_TDES & ( for b5 being Permutation of the carrier of b1 holds
not for b6, b7 being Element of b1 holds
( b5 . b6 = b7 iff ( ( b6 in b4 & b6 = b7 ) or ( not b6 in b4 & ex b8, b9 being Element of b1 st
( b8 in b4 & b9 in b4 & b8,b2 // b9,b6 & b8,b3 // b9,b7 & b6,b7 // b4 ) ) ) ) ) )
proof end;

Lemma32: for b1 being AffinPlane
for b2, b3 being Element of b1
for b4 being Subset of b1
for b5 being Permutation of the carrier of b1 holds
( b2,b3 // b4 & not b2 in b4 & ( for b6, b7 being Element of b1 holds
( b5 . b6 = b7 iff ( ( b6 in b4 & b6 = b7 ) or ( not b6 in b4 & ex b8, b9 being Element of b1 st
( b8 in b4 & b9 in b4 & b8,b2 // b9,b6 & b8,b3 // b9,b7 & b6,b7 // b4 ) ) ) ) ) implies b5 . b2 = b3 )
proof end;

Lemma33: for b1 being AffinPlane
for b2, b3 being Element of b1
for b4 being Subset of b1
for b5 being Permutation of the carrier of b1 holds
( b2,b3 // b4 & not b2 in b4 & ( for b6, b7 being Element of b1 holds
( b5 . b6 = b7 iff ( ( b6 in b4 & b6 = b7 ) or ( not b6 in b4 & ex b8, b9 being Element of b1 st
( b8 in b4 & b9 in b4 & b8,b2 // b9,b6 & b8,b3 // b9,b7 & b6,b7 // b4 ) ) ) ) ) implies for b6 being Element of b1 holds b6,b5 . b6 // b4 )
proof end;

Lemma34: for b1 being AffinPlane
for b2, b3 being Element of b1
for b4 being Subset of b1
for b5 being Permutation of the carrier of b1 holds
( b2,b3 // b4 & not b2 in b4 & b1 satisfies_TDES & ( for b6, b7 being Element of b1 holds
( b5 . b6 = b7 iff ( ( b6 in b4 & b6 = b7 ) or ( not b6 in b4 & ex b8, b9 being Element of b1 st
( b8 in b4 & b9 in b4 & b8,b2 // b9,b6 & b8,b3 // b9,b7 & b6,b7 // b4 ) ) ) ) ) implies b5 is_Col )
proof end;

Lemma35: for b1 being AffinPlane
for b2, b3 being Element of b1
for b4 being Subset of b1
for b5 being Permutation of the carrier of b1 holds
( b2,b3 // b4 & not b2 in b4 & b1 satisfies_TDES & ( for b6, b7 being Element of b1 holds
( b5 . b6 = b7 iff ( ( b6 in b4 & b6 = b7 ) or ( not b6 in b4 & ex b8, b9 being Element of b1 st
( b8 in b4 & b9 in b4 & b8,b2 // b9,b6 & b8,b3 // b9,b7 & b6,b7 // b4 ) ) ) ) ) implies b5 is_Sc b4 )
proof end;

theorem Th8: :: HOMOTHET:8
for b1 being AffinPlane
for b2, b3 being Element of b1
for b4 being Subset of b1 holds
not ( b2,b3 // b4 & not b2 in b4 & b1 satisfies_TDES & ( for b5 being Permutation of the carrier of b1 holds
not ( b5 is_Sc b4 & b5 . b2 = b3 ) ) )
proof end;

theorem Th9: :: HOMOTHET:9
for b1 being AffinPlane holds
( b1 satisfies_TDES iff for b2, b3 being Element of b1
for b4 being Subset of b1 holds
not ( b2,b3 // b4 & not b2 in b4 & ( for b5 being Permutation of the carrier of b1 holds
not ( b5 is_Sc b4 & b5 . b2 = b3 ) ) ) ) by Th6, Th8;