reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));
reserve POS for non empty ParOrtStr;
reserve p,p1,p2,q,q1,r,r1,r2 for Element of AMSpace(V,w,y);

theorem Th23:
  p,q _|_ p1,q1 implies p1,q1 _|_ p,q
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 v1-u1,v-u are_Ort_wrt w,y;
  then u1,v1,u,v are_Ort_wrt w,y;
  hence thesis by Th21;
end;
