reserve n for Nat,
        p,q,u,w for Point of TOP-REAL n,
        S for Subset of TOP-REAL n,
        A, B for convex Subset of TOP-REAL n,
        r for Real;

theorem Th1:
  (1-r)*p + r*q = p + r*(q-p)
  proof
    thus p+r*(q-p) = ((1-r)*p+r*q-p)+p by Lm1
                  .= ((1-r)*p+r*q)-(p-p) by RLVECT_1:29
                  .= ((1-r)*p+r*q)-0.TOP-REAL n by RLVECT_1:15
                  .= (1-r)*p+r*q;
  end;
