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 Th30:
  for W1 being strict Subspace of M holds W1 is Subspace of W3
  implies W1 + (W2 /\ W3) = (W1 + W2) /\ W3
proof
  let W1 be strict Subspace of M;
  assume
A1: W1 is Subspace of W3;
  hence (W1 + W2) /\ W3 = (W1 /\ W3) + (W3 /\ W2) by Lm13,VECTSP_4:29
    .= W1 + (W2 /\ W3) by A1,Th16;
end;
