let V be RealLinearSpace; for u, u1, v, v1 being VECTOR of V
for p, p1, q, q1 being Element of (Lambda (OASpace V)) st p = u & q = v & p1 = u1 & q1 = v1 holds
( p,q // p1,q1 iff ex a, b being Real st
( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) )
let u, u1, v, v1 be VECTOR of V; for p, p1, q, q1 being Element of (Lambda (OASpace V)) st p = u & q = v & p1 = u1 & q1 = v1 holds
( p,q // p1,q1 iff ex a, b being Real st
( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) )
let p, p1, q, q1 be Element of (Lambda (OASpace V)); ( p = u & q = v & p1 = u1 & q1 = v1 implies ( p,q // p1,q1 iff ex a, b being Real st
( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) ) )
assume A1:
( p = u & q = v & p1 = u1 & q1 = v1 )
; ( p,q // p1,q1 iff ex a, b being Real st
( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) )
hereby ( ex a, b being Real st
( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) implies p,q // p1,q1 )
assume
p,
q // p1,
q1
;
ex a, b being Real st
( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) )then
[[p,q],[p1,q1]] in the
CONGR of
(Lambda (OASpace V))
by ANALOAF:def 2;
then
[[u,v],[u1,v1]] in lambda (DirPar V)
by A1, Th17;
hence
ex
a,
b being
Real st
(
a * (v - u) = b * (v1 - u1) & (
a <> 0 or
b <> 0 ) )
by Th15;
verum
end;
given a, b being Real such that A2:
( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) )
; p,q // p1,q1
[[u,v],[u1,v1]] in lambda (DirPar V)
by A2, Th15;
then
[[p,q],[p1,q1]] in the CONGR of (Lambda (OASpace V))
by A1, Th17;
hence
p,q // p1,q1
by ANALOAF:def 2; verum