reserve V for RealLinearSpace;
reserve p,q,u,v,w,y for VECTOR of V;
reserve a,b,c,d for Real;

theorem Th4:
  (a-b)*(u-v) = (b-a)*(v-u)
proof
  thus (a-b)*(u-v)=(-(b-a))*(-(v-u)) by RLVECT_1:33
    .=(b-a)*(v-u) by RLVECT_1:26;
end;
