Article AFF_1, MML version 4.99.1005
:: AFF_1:prednot 1 => AFF_1:pred 1
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3, a4 be Element of the carrier of a1;
pred LIN A2,A3,A4 means
a2,a3 // a2,a4;
end;
:: AFF_1:dfs 1
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3, a4 be Element of the carrier of a1;
To prove
LIN a2,a3,a4
it is sufficient to prove
thus a2,a3 // a2,a4;
:: AFF_1:def 1
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
LIN b2,b3,b4
iff
b2,b3 // b2,b4;
:: AFF_1:th 10
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b2 <> b3;
:: AFF_1:th 11
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 // b3,b2 & b2,b3 // b2,b3;
:: AFF_1:th 12
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 // b4,b4 & b4,b4 // b2,b3;
:: AFF_1:th 13
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 // b4,b5
holds b2,b3 // b5,b4 & b3,b2 // b4,b5 & b3,b2 // b5,b4 & b4,b5 // b2,b3 & b4,b5 // b3,b2 & b5,b4 // b2,b3 & b5,b4 // b3,b2;
:: AFF_1:th 14
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
st b2 <> b3 &
((b2,b3 // b4,b5 implies not b2,b3 // b6,b7) & (b2,b3 // b4,b5 implies not b6,b7 // b2,b3) & (b4,b5 // b2,b3 implies not b6,b7 // b2,b3) implies b4,b5 // b2,b3 & b2,b3 // b6,b7)
holds b4,b5 // b6,b7;
:: AFF_1:th 15
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
st LIN b2,b3,b4
holds LIN b2,b4,b3 & LIN b3,b2,b4 & LIN b3,b4,b2 & LIN b4,b2,b3 & LIN b4,b3,b2;
:: AFF_1:th 16
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
LIN b2,b2,b3 & LIN b2,b3,b3 & LIN b2,b3,b2;
:: AFF_1: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
st b2 <> b3 & LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b3,b6
holds LIN b4,b5,b6;
:: AFF_1: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
st b2 <> b3 & LIN b2,b3,b4 & b2,b3 // b4,b5
holds LIN b2,b3,b5;
:: AFF_1:th 19
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 & LIN b2,b3,b5
holds b2,b3 // b4,b5;
:: AFF_1:th 20
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 b2 <> b3 & LIN b4,b5,b2 & LIN b4,b5,b3 & LIN b2,b3,b6
holds LIN b4,b5,b6;
:: AFF_1:th 21
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
ex b2, b3, b4 being Element of the carrier of b1 st
not LIN b2,b3,b4;
:: AFF_1:th 22
theorem
for b1 being non empty non trivial AffinSpace-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 LIN b2,b3,b4;
:: AFF_1:th 23
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st not LIN b2,b3,b4 & LIN b2,b4,b5 & b3,b4 // b3,b5
holds b4 = b5;
:: AFF_1:funcnot 1 => AFF_1:func 1
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3 be Element of the carrier of a1;
func Line(A2,A3) -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
LIN a2,a3,b1;
end;
:: AFF_1:def 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 holds
b4 = Line(b2,b3)
iff
for b5 being Element of the carrier of b1 holds
b5 in b4
iff
LIN b2,b3,b5;
:: AFF_1:th 25
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
Line(b2,b3) = Line(b3,b2);
:: AFF_1:funcnot 2 => AFF_1:func 2
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3 be Element of the carrier of a1;
redefine func Line(a2,a3) -> Element of bool the carrier of a1;
commutativity;
:: for a1 being non empty non trivial AffinSpace-like AffinStruct
:: for a2, a3 being Element of the carrier of a1 holds
:: Line(a2,a3) = Line(a3,a2);
end;
:: AFF_1:th 26
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2 in Line(b2,b3) & b3 in Line(b2,b3);
:: AFF_1:th 27
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 in Line(b3,b4) & b5 in Line(b3,b4) & b2 <> b5
holds Line(b2,b5) c= Line(b3,b4);
:: AFF_1:th 28
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 in Line(b3,b4) & b5 in Line(b3,b4) & b3 <> b4
holds Line(b3,b4) c= Line(b2,b5);
:: AFF_1:attrnot 1 => AFF_1: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_line means
ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a2 = Line(b1,b2);
end;
:: AFF_1:dfs 3
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_line
it is sufficient to prove
thus ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a2 = Line(b1,b2);
:: AFF_1:def 3
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_line(b1)
iff
ex b3, b4 being Element of the carrier of b1 st
b3 <> b4 & b2 = Line(b3,b4);
:: AFF_1:prednot 2 => AFF_1: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_line for being_line;
end;
:: AFF_1:th 30
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_line(b1) & b5 is being_line(b1) & b2 in b4 & b3 in b4 & b2 in b5 & b3 in b5 & b2 <> b3
holds b4 = b5;
:: AFF_1:th 31
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, b4 being Element of the carrier of b1 st
b3 in b2 & b4 in b2 & b3 <> b4;
:: AFF_1:th 32
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 the carrier of b1 st
b2 <> b4 & b4 in b3;
:: AFF_1:th 33
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
LIN b2,b3,b4
iff
ex b5 being Element of bool the carrier of b1 st
b5 is being_line(b1) & b2 in b5 & b3 in b5 & b4 in b5;
:: AFF_1:prednot 3 => AFF_1:pred 2
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3 be Element of the carrier of a1;
let a4 be Element of bool the carrier of a1;
pred A2,A3 // A4 means
ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a4 = Line(b1,b2) & a2,a3 // b1,b2;
end;
:: AFF_1:dfs 4
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3 be Element of the carrier of a1;
let a4 be Element of bool the carrier of a1;
To prove
a2,a3 // a4
it is sufficient to prove
thus ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a4 = Line(b1,b2) & a2,a3 // b1,b2;
:: AFF_1:def 4
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 holds
b2,b3 // b4
iff
ex b5, b6 being Element of the carrier of b1 st
b5 <> b6 & b4 = Line(b5,b6) & b2,b3 // b5,b6;
:: AFF_1:prednot 4 => AFF_1:pred 3
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
ex b1, b2 being Element of the carrier of a1 st
a2 = Line(b1,b2) & b1 <> b2 & b1,b2 // a3;
end;
:: AFF_1:dfs 5
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 ex b1, b2 being Element of the carrier of a1 st
a2 = Line(b1,b2) & b1 <> b2 & b1,b2 // a3;
:: AFF_1:def 5
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
ex b4, b5 being Element of the carrier of b1 st
b2 = Line(b4,b5) & b4 <> b5 & b4,b5 // b3;
:: AFF_1:th 36
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 in Line(b3,b4) & b3 <> b4
holds b5 in Line(b3,b4)
iff
b3,b4 // b2,b5;
:: AFF_1:th 37
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) & b2 in b4
holds b3 in b4
iff
b2,b3 // b4;
:: AFF_1:th 38
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 holds
b2 <> b3 & b4 = Line(b2,b3)
iff
b4 is being_line(b1) & b2 in b4 & b3 in b4 & b2 <> b3;
:: AFF_1:th 39
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of bool the carrier of b1
st b5 is being_line(b1) & b2 in b5 & b3 in b5 & b2 <> b3 & LIN b2,b3,b4
holds b4 in b5;
:: AFF_1:th 40
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of bool the carrier of b1
st ex b3, b4 being Element of the carrier of b1 st
b3,b4 // b2
holds b2 is being_line(b1);
:: AFF_1:th 41
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 in b6 & b3 in b6 & b6 is being_line(b1) & b2 <> b3
holds b4,b5 // b6
iff
b4,b5 // b2,b3;
:: AFF_1:th 43
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds b2,b3 // Line(b2,b3);
:: AFF_1:th 44
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,b3 // b4
iff
ex b5, b6 being Element of the carrier of b1 st
b5 <> b6 & b5 in b4 & b6 in b4 & b2,b3 // b5,b6;
:: AFF_1:th 45
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_line(b1) & b2,b3 // b6 & b4,b5 // b6
holds b2,b3 // b4,b5;
:: AFF_1:th 46
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 & b2,b3 // b4,b5 & b2 <> b3
holds b4,b5 // b6;
:: AFF_1:th 47
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,b2 // b3;
:: AFF_1:th 48
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
holds b3,b2 // b4;
:: AFF_1:th 49
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 & not b2 in b4
holds not b3 in b4;
:: AFF_1:th 50
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 b2 is being_line(b1) & b3 is being_line(b1);
:: AFF_1:th 51
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
ex b4, b5, b6, b7 being Element of the carrier of b1 st
b4 <> b5 & b6 <> b7 & b4,b5 // b6,b7 & b2 = Line(b4,b5) & b3 = Line(b6,b7);
:: AFF_1:th 52
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 is being_line(b1) & b7 is being_line(b1) & b2 in b6 & b3 in b6 & b4 in b7 & b5 in b7 & b2 <> b3 & b4 <> b5
holds b6 // b7
iff
b2,b3 // b4,b5;
:: AFF_1:th 53
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 b2 in b6 & b3 in b6 & b4 in b7 & b5 in b7 & b6 // b7
holds b2,b3 // b4,b5;
:: AFF_1:th 54
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 in b4 & b3 in b4 & b4 // b5
holds b2,b3 // b5;
:: AFF_1:th 55
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 b2 // b2;
:: AFF_1:th 56
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 b3 // b2;
:: AFF_1:prednot 5 => AFF_1:pred 4
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
let a2, a3 be Element of bool the carrier of a1;
redefine pred a2 // a3;
symmetry;
:: for a1 being non empty non trivial AffinSpace-like AffinStruct
:: for a2, a3 being Element of bool the carrier of a1
:: st a2 // a3
:: holds a3 // a2;
end;
:: AFF_1:th 57
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 & b4 // b5
holds b2,b3 // b5;
:: AFF_1:th 58
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 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_1:th 59
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 & b2 in b3 & b2 in b4
holds b3 = b4;
:: AFF_1:th 60
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of bool the carrier of b1
st b2 in b5 & not b3 in b5 & b3,b4 // b5 & b3 <> b4
holds not LIN b2,b3,b4;
:: AFF_1:th 61
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 being Element of bool the carrier of b1
st b2,b3 // b7 & b4,b5 // b7 & LIN b6,b2,b4 & LIN b6,b3,b5 & b6 in b7 & not b2 in b7 & b2 = b3
holds b4 = b5;
:: AFF_1:th 62
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_line(b1) & b2 in b6 & b3 in b6 & b4 in b6 & b2 <> b3 & b2,b3 // b4,b5
holds b5 in b6;
:: AFF_1:th 63
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 // b4;
:: AFF_1:th 64
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 // b4 & b3 // b5 & b2 in b4 & b2 in b5
holds b4 = b5;
:: AFF_1:th 65
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_line(b1) & b2 in b6 & b3 in b6 & b4 in b6 & b5 in b6
holds b2,b3 // b4,b5;
:: AFF_1:th 66
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) & b2 in b4 & b3 in b4
holds b2,b3 // b4;
:: AFF_1: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, b5 being Element of bool the carrier of b1
st b2,b3 // b4 & b2,b3 // b5 & b2 <> b3
holds b4 // b5;
:: AFF_1:th 68
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 & LIN b2,b3,b5 & LIN b2,b4,b6 & b3,b4 // b5,b6 & b5 = b6
holds b5 = b2 & b6 = b2;
:: AFF_1:th 69
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 & LIN b2,b3,b5 & LIN b2,b4,b6 & b3,b4 // b5,b6 & b5 = b2
holds b6 = b2;
:: AFF_1:th 70
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
st not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b4,b6 & LIN b2,b4,b7 & b3,b4 // b5,b6 & b3,b4 // b5,b7
holds b6 = b7;
:: AFF_1:th 71
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) & b2 in b4 & b3 in b4 & b2 <> b3
holds b4 = Line(b2,b3);
:: AFF_1:th 72
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & not b2 // b3
holds ex b4 being Element of the carrier of b1 st
b4 in b2 & b4 in b3;
:: AFF_1:th 73
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional 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) & not b2,b3 // b4
holds ex b5 being Element of the carrier of b1 st
b5 in b4 & LIN b2,b3,b5;
:: AFF_1:th 74
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st not b2,b3 // b4,b5
holds ex b6 being Element of the carrier of b1 st
LIN b2,b3,b6 & LIN b4,b5,b6;