reserve V for non empty RealLinearSpace;

theorem
  for X be RealLinearSpace, v,w be Element of X,
      v1,w1 be Element of RLSp2RVSp X st
   v=v1 & w=w1 holds v+w=v1+w1 & v-w = v1-w1
proof
  let X be RealLinearSpace, v,w be Element of X,
      v1,w1 be Element of RLSp2RVSp X;
  assume AS: v=v1 & w=w1;
  hence v+w=v1+w1;
  -w = (-1)*w by RLVECT_1:16
      .= (-1.F_Real)*w1 by AS
      .= -w1 by VECTSP_1:14;
  hence v-w = v1-w1 by AS;
end;
