Article AFF_4, MML version 4.99.1005
:: AFF_4:th 1
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st (LIN b2,b3,b4 or LIN b2,b4,b3) & b2 <> b3
holds ex b6 being Element of the carrier of b1 st
LIN b2,b5,b6 & b3,b5 // b4,b6;
:: AFF_4:th 2
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st (b2,b3 // b4 or b3,b2 // b4) & b2 in b4
holds b3 in b4;
:: AFF_4:th 3
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
st (b2,b3 // b4 or b3,b2 // b4) & b4 // b5
holds b2,b3 // b5 & b3,b2 // b5;
:: AFF_4:th 4
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
st (b2,b3 // b6 or b3,b2 // b6) & (b2,b3 // b4,b5 or b4,b5 // b2,b3) & b2 <> b3
holds b4,b5 // b6 & b5,b4 // b6;
:: AFF_4:th 5
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
st (b2,b3 // b4 or b3,b2 // b4) & (b2,b3 // b5 or b3,b2 // b5) & b2 <> b3
holds b4 // b5;
:: AFF_4:th 6
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
st (b2,b3 // b6 or b3,b2 // b6) & (b4,b5 // b6 or b5,b4 // b6)
holds b2,b3 // b4,b5;
:: AFF_4:th 7
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7 being Element of bool the carrier of b1
st (b6 // b7 or b7 // b6) & b2 <> b3 & (b2,b3 // b4,b5 or b4,b5 // b2,b3) & b2 in b6 & b3 in b6 & b4 in b7
holds b5 in b7;
:: AFF_4:th 8
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
for b7, b8 being Element of bool the carrier of b1
st b2 in b7 & b2 in b8 & b3 in b7 & b4 in b7 & b5 in b8 & b6 in b8 & b2 <> b3 & b2 <> b5 & b7 <> b8 & (b3,b5 // b4,b6 or b5,b3 // b6,b4) & b7 is being_line(b1) & b8 is being_line(b1) & b2 = b4
holds b2 = b6;
:: AFF_4:th 9
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
for b7, b8 being Element of bool the carrier of b1
st b2 in b7 & b2 in b8 & b3 in b7 & b4 in b7 & b5 in b8 & b6 in b8 & b2 <> b3 & b2 <> b5 & b7 <> b8 & (b3,b5 // b4,b6 or b5,b3 // b6,b4) & b7 is being_line(b1) & b8 is being_line(b1) & b3 = b4
holds b5 = b6;
:: AFF_4:th 10
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7 being Element of bool the carrier of b1
st (b6 // b7 or b7 // b6) & b2 in b6 & b3 in b6 & b4 in b7 & b5 in b7 & b6 <> b7 & (b2,b4 // b3,b5 or b4,b2 // b5,b3) & b2 = b3
holds b4 = b5;
:: AFF_4:th 11
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of bool the carrier of b1 st
b2 in b4 & b3 in b4 & b4 is being_line(b1);
:: AFF_4:th 12
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1
st b2 is being_line(b1)
holds ex b3 being Element of the carrier of b1 st
not b3 in b2;
:: AFF_4:funcnot 1 => AFF_4:func 1
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3 be Element of bool the carrier of a1;
func Plane(A2,A3) -> Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: ex b2 being Element of the carrier of a1 st
b1,b2 // a2 & b2 in a3};
end;
:: AFF_4:def 1
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1 holds
Plane(b2,b3) = {b4 where b4 is Element of the carrier of b1: ex b5 being Element of the carrier of b1 st
b4,b5 // b2 & b5 in b3};
:: AFF_4:attrnot 1 => AFF_4:attr 1
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is being_plane means
ex b1, b2 being Element of bool the carrier of a1 st
b1 is being_line(a1) & b2 is being_line(a1) & not b1 // b2 & a2 = Plane(b1,b2);
end;
:: AFF_4:dfs 2
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is being_plane
it is sufficient to prove
thus ex b1, b2 being Element of bool the carrier of a1 st
b1 is being_line(a1) & b2 is being_line(a1) & not b1 // b2 & a2 = Plane(b1,b2);
:: AFF_4:def 2
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1 holds
b2 is being_plane(b1)
iff
ex b3, b4 being Element of bool the carrier of b1 st
b3 is being_line(b1) & b4 is being_line(b1) & not b3 // b4 & b2 = Plane(b3,b4);
:: AFF_4:prednot 1 => AFF_4:attr 1
notation
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2 be Element of bool the carrier of a1;
synonym a2 is_plane for being_plane;
end;
:: AFF_4:th 13
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is not being_line(b1)
holds Plane(b2,b3) = {};
:: AFF_4:th 14
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is being_line(b1)
holds b3 c= Plane(b2,b3);
:: AFF_4:th 15
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 // b3
holds Plane(b2,b3) = b3;
:: AFF_4:th 16
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 // b3
holds Plane(b2,b4) = Plane(b3,b4);
:: AFF_4:th 17
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
for b7, b8, b9, b10 being Element of bool the carrier of b1
st b2 in b7 & b3 in b7 & b4 in b7 & b2 in b8 & b5 in b8 & b6 in b8 & not b2 in b9 & not b2 in b10 & b7 <> b8 & b3 in b9 & b5 in b9 & b4 in b10 & b6 in b10 & b7 is being_line(b1) & b8 is being_line(b1) & b9 is being_line(b1) & b10 is being_line(b1) & not b9 // b10
holds ex b11 being Element of the carrier of b1 st
b11 in b9 & b11 in b10;
:: AFF_4:th 18
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8, b9 being Element of bool the carrier of b1
st b2 in b6 & b3 in b6 & b4 in b7 & b5 in b7 & b2 in b8 & b4 in b8 & b3 in b9 & b5 in b9 & b6 <> b7 & b6 // b7 & b8 is being_line(b1) & b9 is being_line(b1) & not b8 // b9
holds ex b10 being Element of the carrier of b1 st
b10 in b8 & b10 in b9;
:: AFF_4:th 19
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st b4 is being_plane(b1) & b2 in b4 & b3 in b4 & b2 <> b3
holds Line(b2,b3) c= b4;
:: AFF_4:th 20
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & not b2 // b3 & not b2 // b4 & b4 c= Plane(b2,b3)
holds Plane(b2,b4) = Plane(b2,b3);
:: AFF_4:th 21
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b4 c= Plane(b2,b3) & not b3 // b4
holds ex b5 being Element of the carrier of b1 st
b5 in b3 & b5 in b4;
:: AFF_4:th 22
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 is being_plane(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b3 c= b2 & b4 c= b2 & not b3 // b4
holds ex b5 being Element of the carrier of b1 st
b5 in b3 & b5 in b4;
:: AFF_4:th 23
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4, b5 being Element of bool the carrier of b1
st b3 is being_plane(b1) & b2 in b3 & b4 c= b3 & b2 in b5 & (b4 // b5 or b5 // b4)
holds b5 c= b3;
:: AFF_4:th 24
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
st b4 is being_plane(b1) & b5 is being_plane(b1) & b2 in b4 & b3 in b4 & b2 in b5 & b3 in b5 & b4 <> b5 & b2 <> b3
holds b4 /\ b5 is being_line(b1);
:: AFF_4:th 25
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Element of bool the carrier of b1
st b5 is being_plane(b1) & b6 is being_plane(b1) & b2 in b5 & b3 in b5 & b4 in b5 & b2 in b6 & b3 in b6 & b4 in b6 & not LIN b2,b3,b4
holds b5 = b6;
:: AFF_4:th 26
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
st b2 is being_plane(b1) & b3 is being_plane(b1) & b4 is being_line(b1) & b5 is being_line(b1) & b4 c= b2 & b5 c= b2 & b4 c= b3 & b5 c= b3 & b4 <> b5
holds b2 = b3;
:: AFF_4:funcnot 2 => AFF_4:func 2
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2 be Element of the carrier of a1;
let a3 be Element of bool the carrier of a1;
assume a3 is being_line(a1);
func A2 * A3 -> Element of bool the carrier of a1 means
a2 in it & a3 // it;
end;
:: AFF_4:def 3
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is being_line(b1)
for b4 being Element of bool the carrier of b1 holds
b4 = b2 * b3
iff
b2 in b4 & b3 // b4;
:: AFF_4:th 27
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is being_line(b1)
holds b2 * b3 is being_line(b1);
:: AFF_4:th 28
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b3 is being_plane(b1) & b4 is being_line(b1) & b2 in b3 & b4 c= b3
holds b2 * b4 c= b3;
:: AFF_4:th 29
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
st b6 is being_plane(b1) & b2 in b6 & b3 in b6 & b4 in b6 & b2,b3 // b4,b5 & b2 <> b3
holds b5 in b6;
:: AFF_4:th 30
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is being_line(b1)
holds b2 in b3
iff
b2 * b3 = b3;
:: AFF_4:th 31
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st b4 is being_line(b1)
holds b2 * b4 = b2 * (b3 * b4);
:: AFF_4:th 32
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b3 // b4
holds b2 * b3 = b2 * b4;
:: AFF_4:prednot 2 => AFF_4:pred 1
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3 be Element of bool the carrier of a1;
pred A2 '||' A3 means
for b1 being Element of the carrier of a1
for b2 being Element of bool the carrier of a1
st b1 in a3 & b2 is being_line(a1) & b2 c= a2
holds b1 * b2 c= a3;
end;
:: AFF_4:dfs 4
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3 be Element of bool the carrier of a1;
To prove
a2 '||' a3
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of bool the carrier of a1
st b1 in a3 & b2 is being_line(a1) & b2 c= a2
holds b1 * b2 c= a3;
:: AFF_4:def 4
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1 holds
b2 '||' b3
iff
for b4 being Element of the carrier of b1
for b5 being Element of bool the carrier of b1
st b4 in b3 & b5 is being_line(b1) & b5 c= b2
holds b4 * b5 c= b3;
:: AFF_4:th 33
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3 &
(b2 is being_line(b1) & b3 is being_line(b1) or b2 is being_plane(b1) & b3 is being_plane(b1))
holds b2 = b3;
:: AFF_4:th 34
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1
st b2 is being_plane(b1)
holds ex b3, b4, b5 being Element of the carrier of b1 st
b3 in b2 & b4 in b2 & b5 in b2 & not LIN b3,b4,b5;
:: AFF_4:th 35
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is being_line(b1) & b3 is being_plane(b1)
holds ex b4 being Element of the carrier of b1 st
b4 in b3 & not b4 in b2;
:: AFF_4:th 36
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is being_line(b1)
holds ex b4 being Element of bool the carrier of b1 st
b2 in b4 & b3 c= b4 & b4 is being_plane(b1);
:: AFF_4:th 37
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of bool the carrier of b1 st
b2 in b5 & b3 in b5 & b4 in b5 & b5 is being_plane(b1);
:: AFF_4:th 38
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b2 in b3 & b2 in b4 & b3 is being_line(b1) & b4 is being_line(b1)
holds ex b5 being Element of bool the carrier of b1 st
b3 c= b5 & b4 c= b5 & b5 is being_plane(b1);
:: AFF_4:th 39
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 // b3
holds ex b4 being Element of bool the carrier of b1 st
b2 c= b4 & b3 c= b4 & b4 is being_plane(b1);
:: AFF_4:th 40
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1)
holds b2 // b3
iff
b2 '||' b3;
:: AFF_4:th 41
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is being_line(b1) & b3 is being_plane(b1)
holds b2 '||' b3
iff
ex b4 being Element of bool the carrier of b1 st
b4 c= b3 & (b2 // b4 or b4 // b2);
:: AFF_4:th 42
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is being_line(b1) & b3 is being_plane(b1) & b2 c= b3
holds b2 '||' b3;
:: AFF_4:th 43
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b3 is being_line(b1) & b4 is being_plane(b1) & b2 in b3 & b2 in b4 & b3 '||' b4
holds b3 c= b4;
:: AFF_4:prednot 3 => AFF_4:pred 2
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3, a4 be Element of bool the carrier of a1;
pred A2,A3,A4 is_coplanar means
ex b1 being Element of bool the carrier of a1 st
a2 c= b1 & a3 c= b1 & a4 c= b1 & b1 is being_plane(a1);
end;
:: AFF_4:dfs 5
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3, a4 be Element of bool the carrier of a1;
To prove
a2,a3,a4 is_coplanar
it is sufficient to prove
thus ex b1 being Element of bool the carrier of a1 st
a2 c= b1 & a3 c= b1 & a4 c= b1 & b1 is being_plane(a1);
:: AFF_4:def 5
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1 holds
b2,b3,b4 is_coplanar
iff
ex b5 being Element of bool the carrier of b1 st
b2 c= b5 & b3 c= b5 & b4 c= b5 & b5 is being_plane(b1);
:: AFF_4:th 44
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2,b3,b4 is_coplanar
holds b2,b4,b3 is_coplanar & b3,b2,b4 is_coplanar & b3,b4,b2 is_coplanar & b4,b2,b3 is_coplanar & b4,b3,b2 is_coplanar;
:: AFF_4:th 45
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b5 is being_line(b1) & b4,b5,b3 is_coplanar & b4,b5,b2 is_coplanar & b4 <> b5
holds b4,b3,b2 is_coplanar;
:: AFF_4:th 46
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_plane(b1) & b2 c= b4 & b3 c= b4 & b2 <> b3
holds b2,b3,b5 is_coplanar
iff
b5 c= b4;
:: AFF_4:th 47
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b2 in b3 & b2 in b4 & b3 is being_line(b1) & b4 is being_line(b1)
holds b3,b4,b4 is_coplanar & b4,b3,b4 is_coplanar & b4,b4,b3 is_coplanar;
:: AFF_4:th 48
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1
st b1 is not non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
b2 is being_plane(b1)
holds ex b3 being Element of the carrier of b1 st
not b3 in b2;
:: AFF_4:th 49
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
for b9, b10, b11 being Element of bool the carrier of b1
st b1 is not non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
b2 in b9 &
b2 in b10 &
b2 in b11 &
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b3 in b9 &
b6 in b9 &
b4 in b10 &
b7 in b10 &
b5 in b11 &
b8 in b11 &
b9 is being_line(b1) &
b10 is being_line(b1) &
b11 is being_line(b1) &
b9 <> b10 &
b9 <> b11 &
b3,b4 // b6,b7 &
b3,b5 // b6,b8
holds b4,b5 // b7,b8;
:: AFF_4:th 50
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
st b1 is not non empty non trivial AffinSpace-like 2-dimensional AffinStruct
holds b1 is Desarguesian;
:: AFF_4:th 51
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
for b8, b9, b10 being Element of bool the carrier of b1
st b1 is not non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
b8 // b9 &
b8 // b10 &
b2 in b8 &
b3 in b8 &
b4 in b9 &
b5 in b9 &
b6 in b10 &
b7 in b10 &
b8 is being_line(b1) &
b9 is being_line(b1) &
b10 is being_line(b1) &
b8 <> b9 &
b8 <> b10 &
b2,b4 // b3,b5 &
b2,b6 // b3,b7
holds b4,b6 // b5,b7;
:: AFF_4:th 52
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
st b1 is not non empty non trivial AffinSpace-like 2-dimensional AffinStruct
holds b1 is translational;
:: AFF_4:th 53
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b1 is non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
not LIN b2,b3,b4
holds ex b7 being Element of the carrier of b1 st
b2,b4 // b5,b7 & b3,b4 // b6,b7;
:: AFF_4:th 54
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st not LIN b2,b3,b4 & b5 <> b6 & b2,b3 // b5,b6
holds ex b7 being Element of the carrier of b1 st
b2,b4 // b5,b7 & b3,b4 // b6,b7;
:: AFF_4:th 55
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is being_plane(b1) & b3 is being_plane(b1)
holds b2 '||' b3
iff
ex b4, b5, b6, b7 being Element of bool the carrier of b1 st
not b4 // b5 & b4 c= b2 & b5 c= b2 & b6 c= b3 & b7 c= b3 & (b4 // b6 or b6 // b4) & (b5 // b7 or b7 // b5);
:: AFF_4:th 56
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 // b3 & b3 '||' b4
holds b2 '||' b4;
:: AFF_4:th 57
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1
st b2 is being_plane(b1)
holds b2 '||' b2;
:: AFF_4:th 58
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is being_plane(b1) & b3 is being_plane(b1) & b2 '||' b3
holds b3 '||' b2;
:: AFF_4:th 59
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1
st b2 is being_plane(b1)
holds b2 <> {};
:: AFF_4:th 60
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 '||' b3 & b3 '||' b4 & b3 <> {}
holds b2 '||' b4;
:: AFF_4:th 61
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 is being_plane(b1) &
b3 is being_plane(b1) &
b4 is being_plane(b1) &
((b2 '||' b3 implies not b3 '||' b4) & (b2 '||' b3 implies not b4 '||' b3) & (b3 '||' b2 implies not b3 '||' b4) implies b3 '||' b2 & b4 '||' b3)
holds b2 '||' b4;
:: AFF_4:th 62
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b3 is being_plane(b1) & b4 is being_plane(b1) & b2 in b3 & b2 in b4 & b3 '||' b4
holds b3 = b4;
:: AFF_4:th 63
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8 being Element of bool the carrier of b1
st b6 is being_plane(b1) & b7 is being_plane(b1) & b8 is being_plane(b1) & b6 '||' b7 & b6 <> b7 & b2 in b6 /\ b8 & b3 in b6 /\ b8 & b4 in b7 /\ b8 & b5 in b7 /\ b8
holds b2,b3 // b4,b5;
:: AFF_4:th 64
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8 being Element of bool the carrier of b1
st b6 is being_plane(b1) & b7 is being_plane(b1) & b8 is being_plane(b1) & b6 '||' b7 & b2 in b6 /\ b8 & b3 in b6 /\ b8 & b4 in b7 /\ b8 & b5 in b7 /\ b8 & b6 <> b7 & b2 <> b3 & b4 <> b5
holds b6 /\ b8 // b7 /\ b8;
:: AFF_4:th 65
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is being_plane(b1)
holds ex b4 being Element of bool the carrier of b1 st
b2 in b4 & b3 '||' b4 & b4 is being_plane(b1);
:: AFF_4:funcnot 3 => AFF_4:func 3
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2 be Element of the carrier of a1;
let a3 be Element of bool the carrier of a1;
assume a3 is being_plane(a1);
func A2 + A3 -> Element of bool the carrier of a1 means
a2 in it & a3 '||' it & it is being_plane(a1);
end;
:: AFF_4:def 6
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is being_plane(b1)
for b4 being Element of bool the carrier of b1 holds
b4 = b2 + b3
iff
b2 in b4 & b3 '||' b4 & b4 is being_plane(b1);
:: AFF_4:th 66
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is being_plane(b1)
holds b2 in b3
iff
b2 + b3 = b3;
:: AFF_4:th 67
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st b4 is being_plane(b1)
holds b2 + b4 = b2 + (b3 + b4);
:: AFF_4:th 68
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b3 is being_line(b1) & b4 is being_plane(b1) & b3 '||' b4
holds b2 * b3 c= b2 + b4;
:: AFF_4:th 69
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b3 is being_plane(b1) & b4 is being_plane(b1) & b3 '||' b4
holds b2 + b3 = b2 + b4;