reserve V for RealLinearSpace;
reserve W,W1,W2,W3 for Subspace of V;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a,a1,a2 for Real;
reserve X,Y,x,y,y1,y2 for set;

theorem
  for W2,W3 being strict Subspace of V holds W1 is Subspace of W2
  implies W1 + W3 is Subspace of W2 + W3
proof
  let W2,W3 be strict Subspace of V;
  assume
A1: W1 is Subspace of W2;
  (W1 + W3) + (W2 + W3) = (W1 + W3) + (W3 + W2) by Lm1
    .= ((W1 + W3) + W3) + W2 by Th6
    .= (W1 + (W3 + W3)) + W2 by Th6
    .= (W1 + W3) + W2 by Lm3
    .= W1 + (W3 + W2) by Th6
    .= W1 + (W2 + W3) by Lm1
    .= (W1 + W2) + W3 by Th6
    .= W2 + W3 by A1,Th8;
  hence thesis by Th8;
end;
