Article DIRAF, MML version 4.99.1005
:: DIRAF:funcnot 1 => DIRAF:func 1
definition
let a1 be non empty set;
let a2 be Relation of [:a1,a1:],[:a1,a1:];
func lambda A2 -> Relation of [:a1,a1:],[:a1,a1:] means
for b1, b2, b3, b4 being Element of a1 holds
[[b1,b2],[b3,b4]] in it
iff
([[b1,b2],[b3,b4]] in a2 or [[b1,b2],[b4,b3]] in a2);
end;
:: DIRAF:def 1
theorem
for b1 being non empty set
for b2, b3 being Relation of [:b1,b1:],[:b1,b1:] holds
b3 = lambda b2
iff
for b4, b5, b6, b7 being Element of b1 holds
[[b4,b5],[b6,b7]] in b3
iff
([[b4,b5],[b6,b7]] in b2 or [[b4,b5],[b7,b6]] in b2);
:: DIRAF:funcnot 2 => DIRAF:func 2
definition
let a1 be non empty AffinStruct;
func Lambda A1 -> strict AffinStruct equals
AffinStruct(#the carrier of a1,lambda the CONGR of a1#);
end;
:: DIRAF:def 2
theorem
for b1 being non empty AffinStruct holds
Lambda b1 = AffinStruct(#the carrier of b1,lambda the CONGR of b1#);
:: DIRAF:funcreg 1
registration
let a1 be non empty AffinStruct;
cluster Lambda a1 -> non empty strict;
end;
:: DIRAF:th 4
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 // b2,b3;
:: DIRAF:th 5
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 // b4,b5
holds b3,b2 // b5,b4 & b4,b5 // b2,b3 & b5,b4 // b3,b2;
:: DIRAF:th 6
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 7
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b2 // b3,b4 & b3,b4 // b2,b2;
:: DIRAF:th 8
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 // b4,b5 & b2,b3 // b5,b4 & b2 <> b3
holds b4 = b5;
:: DIRAF:th 9
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 // b2,b4
iff
(b2,b3 // b3,b4 or b2,b4 // b4,b3);
:: DIRAF:prednot 1 => DIRAF:pred 1
definition
let a1 be non empty AffinStruct;
let a2, a3, a4 be Element of the carrier of a1;
pred Mid A2,A3,A4 means
a2,a3 // a3,a4;
end;
:: DIRAF:dfs 3
definiens
let a1 be non empty AffinStruct;
let a2, a3, a4 be Element of the carrier of a1;
To prove
Mid a2,a3,a4
it is sufficient to prove
thus a2,a3 // a3,a4;
:: DIRAF:def 3
theorem
for b1 being non empty AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
Mid b2,b3,b4
iff
b2,b3 // b3,b4;
:: DIRAF:th 11
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 // b2,b4
iff
(Mid b2,b3,b4 or Mid b2,b4,b3);
:: DIRAF:th 12
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
st Mid b2,b3,b2
holds b2 = b3;
:: DIRAF:th 13
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
st Mid b2,b3,b4
holds Mid b4,b3,b2;
:: DIRAF:th 14
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
Mid b2,b2,b3 & Mid b2,b3,b3;
:: DIRAF:th 15
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Mid b2,b3,b4 & Mid b2,b4,b5
holds Mid b3,b4,b5;
:: DIRAF:th 16
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <> b3 & Mid b4,b2,b3 & Mid b2,b3,b5
holds Mid b4,b3,b5;
:: DIRAF:th 17
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
Mid b2,b3,b4 & b3 <> b4;
:: DIRAF:th 18
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
st Mid b2,b3,b4 & Mid b3,b2,b4
holds b2 = b3;
:: DIRAF:th 19
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <> b3 & Mid b2,b3,b4 & Mid b2,b3,b5 & not Mid b3,b4,b5
holds Mid b3,b5,b4;
:: DIRAF:th 20
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <> b3 & Mid b2,b3,b4 & Mid b2,b3,b5 & not Mid b2,b4,b5
holds Mid b2,b5,b4;
:: DIRAF:th 21
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Mid b2,b3,b4 & Mid b2,b5,b4 & not Mid b2,b3,b5
holds Mid b2,b5,b3;
:: DIRAF:prednot 2 => DIRAF:pred 2
definition
let a1 be non empty AffinStruct;
let a2, a3, a4, a5 be Element of the carrier of a1;
pred A2,A3 '||' A4,A5 means
(not a2,a3 // a4,a5) implies a2,a3 // a5,a4;
end;
:: DIRAF:dfs 4
definiens
let a1 be non empty AffinStruct;
let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
a2,a3 '||' a4,a5
it is sufficient to prove
thus (not a2,a3 // a4,a5) implies a2,a3 // a5,a4;
:: DIRAF:def 4
theorem
for b1 being non empty AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2,b3 '||' b4,b5
iff
(b2,b3 // b4,b5 or b2,b3 // b5,b4);
:: DIRAF:th 23
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2,b3 '||' b4,b5
iff
[[b2,b3],[b4,b5]] in lambda the CONGR of b1;
:: DIRAF:th 24
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 '||' b3,b2 & b2,b3 '||' b2,b3;
:: DIRAF:th 25
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 '||' b4,b4 & b4,b4 '||' b2,b3;
:: DIRAF:th 26
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3 '||' b2,b4
holds b3,b2 '||' b3,b4;
:: DIRAF:th 27
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 28
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 29
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
ex b2, b3, b4 being Element of the carrier of b1 st
not b2,b3 '||' b2,b4;
:: DIRAF:th 30
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
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 & b4 <> b5;
:: DIRAF:th 31
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
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;
:: DIRAF:th 32
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 '||' b3,b4 & b3 <> b2
holds ex b6 being Element of the carrier of b1 st
b5,b3 '||' b3,b6 & b5,b2 '||' b4,b6;
:: DIRAF:prednot 3 => DIRAF:pred 3
definition
let a1 be non empty AffinStruct;
let a2, a3, a4 be Element of the carrier of a1;
pred LIN A2,A3,A4 means
a2,a3 '||' a2,a4;
end;
:: DIRAF:dfs 5
definiens
let a1 be non empty 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;
:: DIRAF:def 5
theorem
for b1 being non empty AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
LIN b2,b3,b4
iff
b2,b3 '||' b2,b4;
:: DIRAF:prednot 4 => DIRAF:pred 3
notation
let a1 be non empty AffinStruct;
let a2, a3, a4 be Element of the carrier of a1;
synonym a2,a3,a4 is_collinear for LIN a2,a3,a4;
end;
:: DIRAF:th 34
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
st Mid b2,b3,b4
holds LIN b2,b3,b4;
:: DIRAF:th 35
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
st LIN b2,b3,b4 & not Mid b2,b3,b4 & not Mid b3,b2,b4
holds Mid b2,b4,b3;
:: DIRAF:th 36
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 37
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 38
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 39
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 40
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 41
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 42
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
ex b2, b3, b4 being Element of the carrier of b1 st
not LIN b2,b3,b4;
:: DIRAF:th 43
theorem
for b1 being non empty non trivial OAffinSpace-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;
:: DIRAF:th 45
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being non empty AffinStruct
st b2 = Lambda b1
for b3, b4, b5, b6 being Element of the carrier of b1
for b7, b8, b9, b10 being Element of the carrier of b2
st b3 = b7 & b4 = b8 & b5 = b9 & b6 = b10
holds b7,b8 // b9,b10
iff
b3,b4 '||' b5,b6;
:: DIRAF:th 46
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being non empty AffinStruct
st b2 = Lambda b1
holds (ex b3, b4 being Element of the carrier of b2 st
b3 <> b4) &
(for b3, b4, b5, b6, b7, b8 being Element of the carrier of b2 holds
b3,b4 // b4,b3 &
b3,b4 // b5,b5 &
(b3 <> b4 & b3,b4 // b5,b6 & b3,b4 // b7,b8 implies b5,b6 // b7,b8) &
(b3,b4 // b3,b5 implies b4,b3 // b4,b5)) &
(ex b3, b4, b5 being Element of the carrier of b2 st
not b3,b4 // b3,b5) &
(for b3, b4, b5 being Element of the carrier of b2 holds
ex b6 being Element of the carrier of b2 st
b3,b5 // b4,b6 & b4 <> b6) &
(for b3, b4, b5 being Element of the carrier of b2 holds
ex b6 being Element of the carrier of b2 st
b3,b4 // b5,b6 & b3,b5 // b4,b6) &
(for b3, b4, b5, b6 being Element of the carrier of b2
st b5,b3 // b3,b6 & b3 <> b5
holds ex b7 being Element of the carrier of b2 st
b4,b3 // b3,b7 & b4,b5 // b6,b7);
:: DIRAF:attrnot 1 => DIRAF:attr 1
definition
let a1 be non empty AffinStruct;
attr a1 is AffinSpace-like means
(for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1 holds
b1,b2 // b2,b1 &
b1,b2 // b3,b3 &
(b1 <> b2 & b1,b2 // b3,b4 & b1,b2 // b5,b6 implies b3,b4 // b5,b6) &
(b1,b2 // b1,b3 implies 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,b3 // b2,b4 & b2 <> b4) &
(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, b3, b4 being Element of the carrier of a1
st b3,b1 // b1,b4 & b1 <> b3
holds ex b5 being Element of the carrier of a1 st
b2,b1 // b1,b5 & b2,b3 // b4,b5);
end;
:: DIRAF:dfs 6
definiens
let a1 be non empty AffinStruct;
To prove
a1 is AffinSpace-like
it is sufficient to prove
thus (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1 holds
b1,b2 // b2,b1 &
b1,b2 // b3,b3 &
(b1 <> b2 & b1,b2 // b3,b4 & b1,b2 // b5,b6 implies b3,b4 // b5,b6) &
(b1,b2 // b1,b3 implies 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,b3 // b2,b4 & b2 <> b4) &
(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, b3, b4 being Element of the carrier of a1
st b3,b1 // b1,b4 & b1 <> b3
holds ex b5 being Element of the carrier of a1 st
b2,b1 // b1,b5 & b2,b3 // b4,b5);
:: DIRAF:def 7
theorem
for b1 being non empty AffinStruct holds
b1 is AffinSpace-like
iff
(for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
b2,b3 // b3,b2 &
b2,b3 // b4,b4 &
(b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7 implies b4,b5 // b6,b7) &
(b2,b3 // b2,b4 implies 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,b4 // b3,b5 & b3 <> b5) &
(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, b4, b5 being Element of the carrier of b1
st b4,b2 // b2,b5 & b2 <> b4
holds ex b6 being Element of the carrier of b1 st
b3,b2 // b2,b6 & b3,b4 // b5,b6);
:: DIRAF:exreg 1
registration
cluster non empty non trivial strict AffinSpace-like AffinStruct;
end;
:: DIRAF:modenot 1
definition
mode AffinSpace is non empty non trivial AffinSpace-like AffinStruct;
end;
:: DIRAF:th 47
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
(ex b2, b3 being Element of the carrier of b1 st
b2 <> b3) &
(for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
b2,b3 // b3,b2 &
b2,b3 // b4,b4 &
(b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7 implies b4,b5 // b6,b7) &
(b2,b3 // b2,b4 implies 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,b4 // b3,b5 & b3 <> b5) &
(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, b4, b5 being Element of the carrier of b1
st b4,b2 // b2,b5 & b2 <> b4
holds ex b6 being Element of the carrier of b1 st
b3,b2 // b2,b6 & b3,b4 // b5,b6);
:: DIRAF:th 48
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
Lambda b1 is non empty non trivial AffinSpace-like AffinStruct;
:: DIRAF:th 49
theorem
for b1 being non empty AffinStruct holds
(ex b2, b3 being Element of the carrier of b1 st
b2 <> b3) &
(for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
b2,b3 // b3,b2 &
b2,b3 // b4,b4 &
(b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7 implies b4,b5 // b6,b7) &
(b2,b3 // b2,b4 implies 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,b4 // b3,b5 & b3 <> b5) &
(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, b4, b5 being Element of the carrier of b1
st b4,b2 // b2,b5 & b2 <> b4
holds ex b6 being Element of the carrier of b1 st
b3,b2 // b2,b6 & b3,b4 // b5,b6)
iff
b1 is non empty non trivial AffinSpace-like AffinStruct;
:: DIRAF:th 50
theorem
for b1 being non empty non trivial OAffinSpace-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
b2,b3 '||' b2,b6 & b4,b5 '||' b4,b6;
:: DIRAF:th 51
theorem
for b1 being non empty AffinStruct
for b2 being non empty non trivial OAffinSpace-like 2-dimensional AffinStruct
st b1 = Lambda b2
for b3, b4, b5, b6 being Element of the carrier of b1
st not b3,b4 // b5,b6
holds ex b7 being Element of the carrier of b1 st
b3,b4 // b3,b7 & b5,b6 // b5,b7;
:: DIRAF:attrnot 2 => DIRAF:attr 2
definition
let a1 be non empty AffinStruct;
attr a1 is 2-dimensional means
for b1, b2, b3, b4 being Element of the carrier of a1
st not b1,b2 // b3,b4
holds ex b5 being Element of the carrier of a1 st
b1,b2 // b1,b5 & b3,b4 // b3,b5;
end;
:: DIRAF:dfs 7
definiens
let a1 be non empty AffinStruct;
To prove
a1 is 2-dimensional
it is sufficient to prove
thus for b1, b2, b3, b4 being Element of the carrier of a1
st not b1,b2 // b3,b4
holds ex b5 being Element of the carrier of a1 st
b1,b2 // b1,b5 & b3,b4 // b3,b5;
:: DIRAF:def 8
theorem
for b1 being non empty AffinStruct holds
b1 is 2-dimensional
iff
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
b2,b3 // b2,b6 & b4,b5 // b4,b6;
:: DIRAF:exreg 2
registration
cluster non empty non trivial strict AffinSpace-like 2-dimensional AffinStruct;
end;
:: DIRAF:modenot 2
definition
mode AffinPlane is non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
end;
:: DIRAF:th 53
theorem
for b1 being non empty non trivial OAffinSpace-like 2-dimensional AffinStruct holds
Lambda b1 is non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
:: DIRAF:th 54
theorem
for b1 being non empty AffinStruct holds
b1 is non empty non trivial AffinSpace-like 2-dimensional AffinStruct
iff
(ex b2, b3 being Element of the carrier of b1 st
b2 <> b3) &
(for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
b2,b3 // b3,b2 &
b2,b3 // b4,b4 &
(b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7 implies b4,b5 // b6,b7) &
(b2,b3 // b2,b4 implies 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,b4 // b3,b5 & b3 <> b5) &
(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, b4, b5 being Element of the carrier of b1
st b4,b2 // b2,b5 & b2 <> b4
holds ex b6 being Element of the carrier of b1 st
b3,b2 // b2,b6 & b3,b4 // b5,b6) &
(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
b2,b3 // b2,b6 & b4,b5 // b4,b6);