let AS be AffinSpace; for a, b, c, d being Element of AS
for A, C being being_line Subset of AS st a in A & b in A & c in C & d in C & a <> b & c <> d holds
( A // C iff a,b // c,d )
let a, b, c, d be Element of AS; for A, C being being_line Subset of AS st a in A & b in A & c in C & d in C & a <> b & c <> d holds
( A // C iff a,b // c,d )
let A, C be being_line Subset of AS; ( a in A & b in A & c in C & d in C & a <> b & c <> d implies ( A // C iff a,b // c,d ) )
assume that
A3:
a in A
and
A4:
b in A
and
A5:
c in C
and
A6:
d in C
and
A7:
a <> b
and
A8:
c <> d
; ( A // C iff a,b // c,d )
thus
( A // C implies a,b // c,d )
( a,b // c,d implies A // C )proof
assume
A // C
;
a,b // c,d
then consider p,
q,
r,
s being
Element of
AS such that A10:
p <> q
and A11:
r <> s
and A12:
p,
q // r,
s
and A13:
A = Line (
p,
q)
and A14:
C = Line (
r,
s)
by Th51;
p,
q // a,
b
by A3, A4, A10, A13, Th36;
then A15:
a,
b // r,
s
by A10, A12, Th14;
r,
s // c,
d
by A5, A6, A11, A14, Th36;
hence
a,
b // c,
d
by A11, A15, Th14;
verum
end;
A16:
C = Line (c,d)
by A5, A6, A8, Lm6;
assume A17:
a,b // c,d
; A // C
A = Line (a,b)
by A3, A4, A7, Lm6;
hence
A // C
by A7, A8, A17, A16, Th51; verum