let V be non trivial RealLinearSpace; for p, q, r being Element of (ProjectiveSpace V) holds
( p,q,r are_collinear iff ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & ( u = v or u = w or v = w or {u,v,w} is linearly-dependent ) ) )
let p, q, r be Element of (ProjectiveSpace V); ( p,q,r are_collinear iff ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & ( u = v or u = w or v = w or {u,v,w} is linearly-dependent ) ) )
hereby ( ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & ( u = v or u = w or v = w or {u,v,w} is linearly-dependent ) ) implies p,q,r are_collinear )
assume
p,
q,
r are_collinear
;
ex u, v, w being Element of V st
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & ( u = v or u = w or v = w or {u,v,w} is linearly-dependent ) )then consider u,
v,
w being
Element of
V such that A1:
(
p = Dir u &
q = Dir v &
r = Dir w & not
u is
zero & not
v is
zero & not
w is
zero &
u,
v,
w are_LinDep )
by ANPROJ_2:23;
(
p = Dir u &
q = Dir v &
r = Dir w & not
u is
zero & not
v is
zero & not
w is
zero & (
u = v or
u = w or
v = w or
{u,v,w} is
linearly-dependent ) )
by A1, Th8;
hence
ex
u,
v,
w being
Element of
V st
(
p = Dir u &
q = Dir v &
r = Dir w & not
u is
zero & not
v is
zero & not
w is
zero & (
u = v or
u = w or
v = w or
{u,v,w} is
linearly-dependent ) )
;
verum
end;
given u, v, w being Element of V such that A2:
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & ( u = v or u = w or v = w or {u,v,w} is linearly-dependent ) )
; p,q,r are_collinear
( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep )
by Th8, A2;
hence
p,q,r are_collinear
by ANPROJ_2:23; verum