let V be RealLinearSpace; :: thesis: for u, u1, v, v1, w, x, y being VECTOR of V st Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,v1,w are_COrtm_wrt x,y holds
u,v,u1,w are_COrtm_wrt x,y
let u, u1, v, v1, w, x, y be VECTOR of V; :: thesis: ( Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,v1,w are_COrtm_wrt x,y implies u,v,u1,w are_COrtm_wrt x,y )
assume A1: Gen x,y ; :: thesis: ( not u,v,u1,v1 are_COrtm_wrt x,y or not u,v,v1,w are_COrtm_wrt x,y or u,v,u1,w are_COrtm_wrt x,y )
assume that
A2: u,v,u1,v1 are_COrtm_wrt x,y and
A3: u,v,v1,w are_COrtm_wrt x,y ; :: thesis: u,v,u1,w are_COrtm_wrt x,y
A4: Ortm (x,y,u), Ortm (x,y,v) // u1,v1 by A2;
A5: Ortm (x,y,u), Ortm (x,y,v) // v1,w by A3;
A6: u1,v1 // Ortm (x,y,u), Ortm (x,y,v) by A4, ANALOAF:12;

A7: now :: thesis: ( u <> v implies u,v,u1,w are_COrtm_wrt x,y )

assume u <> v ; :: thesis: u,v,u1,w are_COrtm_wrt x,y
then u1,v1 // v1,w by A1, A4, A5, Th6, ANALOAF:11;
then A8: u1,v1 // u1,w by ANALOAF:13;
( u1 <> v1 implies u,v,u1,w are_COrtm_wrt x,y ) by A6, A8, ANALOAF:11;
hence u,v,u1,w are_COrtm_wrt x,y by A3; :: thesis: verum

end;

( u = v implies u,v,u1,w are_COrtm_wrt x,y ) by ANALOAF:9;
hence u,v,u1,w are_COrtm_wrt x,y by A7; :: thesis: verum