reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
reserve M for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF;
reserve W,W1,W2,W3 for Subspace of M;
reserve u,u1,u2,v,v1,v2 for Element of M;
reserve X,Y for set, x,y,y1,y2 for object;

theorem
  for W2,W3 being strict Subspace of M holds W1 is Subspace of W2
  implies W1 + W3 is Subspace of W2 + W3
proof
  let W2,W3 be strict Subspace of M;
  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;
