theorem Th24:
  p,q _|_ p1,q1 implies p,q _|_ q1,p1
proof
  reconsider u=p,v=q,u1=p1,v1=q1 as Element of V;
  assume p,q _|_ p1,q1;
  then u,v,u1,v1 are_Ort_wrt w,y by Th21;
  then v-u,v1-u1 are_Ort_wrt w,y;
  then
A1: v-u,(-1)*(v1-u1) are_Ort_wrt w,y by Th7;
  (-1)*(v1-u1) = -(v1-u1) by RLVECT_1:16
    .= u1-v1 by RLVECT_1:33;
  then u,v,v1,u1 are_Ort_wrt w,y by A1;
  hence thesis by Th21;
end;
