reserve V for non empty RealLinearSpace;

theorem
  for X be VectSp of F_Real, v,w be Element of X,
      v1,w1 be Element of RVSp2RLSp X st
    v=v1 & w=w1 holds v+w = v1+w1 & v-w = v1-w1
proof
  let X be VectSp of F_Real, v,w be Element of X,
      v1,w1 be Element of RVSp2RLSp X;
  assume AS: v=v1 & w=w1;
  -w1 = (-1)*w1 by RLVECT_1:16
       .= (-1.F_Real)*w by AS
       .= -w by VECTSP_1:14;
  hence v+w = v1+w1 & v-w = v1-w1 by AS;
end;
