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,u1,w are_COrtm_wrt x,y & not u,v,v1,w are_COrtm_wrt x,y holds
u,v,w,v1 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,u1,w are_COrtm_wrt x,y & not u,v,v1,w are_COrtm_wrt x,y implies u,v,w,v1 are_COrtm_wrt x,y )
assume that
A1: Gen x,y and
A2: u,v,u1,v1 are_COrtm_wrt x,y and
A3: u,v,u1,w are_COrtm_wrt x,y ; :: thesis: ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 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) // u1,w by A3;

A6: now :: thesis: ( u <> v & u1 <> v1 & not u,v,v1,w are_COrtm_wrt x,y implies u,v,w,v1 are_COrtm_wrt x,y )

assume that
A7: u <> v and
A8: u1 <> v1 ; :: thesis: ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y )
u1,v1 // u1,w by A1, A4, A5, A7, Th6, ANALOAF:11;
then A9: ( u1,v1 // v1,w or u1,w // w,v1 ) by ANALOAF:14;

now :: thesis: ( u1 <> w & not u,v,v1,w are_COrtm_wrt x,y implies u,v,w,v1 are_COrtm_wrt x,y )

assume A10: u1 <> w ; :: thesis: ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y )
A11: u1,v1 // Ortm (x,y,u), Ortm (x,y,v) by A4, ANALOAF:12;
u1,w // Ortm (x,y,u), Ortm (x,y,v) by A5, ANALOAF:12;
then ( Ortm (x,y,u), Ortm (x,y,v) // v1,w or Ortm (x,y,u), Ortm (x,y,v) // w,v1 ) by A8, A9, A10, A11, ANALOAF:11;
hence ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y ) ; :: thesis: verum

end;

hence ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y ) by A2; :: thesis: verum

end;

( u = v & not u,v,v1,w are_COrtm_wrt x,y implies u,v,w,v1 are_COrtm_wrt x,y ) by ANALOAF:9;
hence ( u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y ) by A3, A6; :: thesis: verum