let V be RealLinearSpace; :: thesis: for u, u1, v, v1, w, y being VECTOR of V st u,v,u1,v1 are_Ort_wrt w,y holds
u,v,v1,u1 are_Ort_wrt w,y
let u, u1, v, v1, w, y be VECTOR of V; :: thesis: ( u,v,u1,v1 are_Ort_wrt w,y implies u,v,v1,u1 are_Ort_wrt w,y )
assume u,v,u1,v1 are_Ort_wrt w,y ; :: thesis: u,v,v1,u1 are_Ort_wrt w,y
then v - u,v1 - u1 are_Ort_wrt w,y by ANALMETR:def 3;
then A1: v - u,(- 1) * (v1 - u1) are_Ort_wrt w,y by ANALMETR:7;
(- 1) * (v1 - u1) = - (v1 - u1) by RLVECT_1:16
.= u1 - v1 by RLVECT_1:33 ;
hence u,v,v1,u1 are_Ort_wrt w,y by A1, ANALMETR:def 3; :: thesis: verum