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;