reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;

theorem Th30:
  for W1,W2 being strict Subspace of V st (for v holds v in W1 iff
  v in W2) holds W1 = W2
proof
  let W1,W2 be strict Subspace of V;
  assume
A1: for v holds v in W1 iff v in W2;
  for x being object holds x in the carrier of W1 iff x in the carrier of W2
  proof let x be object;
    thus x in the carrier of W1 implies x in the carrier of W2
    proof
      assume
A2:   x in the carrier of W1;
      the carrier of W1 c= the carrier of V by Def2;
      then reconsider v = x as Element of V by A2;
      v in W1 by A2;
      then v in W2 by A1;
      hence thesis;
    end;
    assume
A3: x in the carrier of W2;
    the carrier of W2 c= the carrier of V by Def2;
    then reconsider v = x as Element of V by A3;
    v in W2 by A3;
    then v in W1 by A1;
    hence thesis;
  end;
  then the carrier of W1 = the carrier of W2 by TARSKI:2;
  hence thesis by Th29;
end;
