let V be non trivial RealLinearSpace; :: thesis: 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); :: thesis: ( 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 :: thesis: ( 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 ; :: thesis: 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 ) ) ; :: thesis: 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 ) ) ; :: thesis: 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; :: thesis: verum