reserve C for non empty set;
reserve GF for Field,
        V for VectSp of GF,
        v,u for Element of V,
        W for Subset of V;
reserve f,f1,f2,f3 for PartFunc of C,V;

theorem MEASURE624:
  for x being Element of V, A being Subset of V st A = the carrier of V
  holds x ++ A = A
proof
  let x be Element of V, A be Subset of V;
  assume
P0: A = the carrier of V;
  for y being object holds y in x ++ A iff y in A
  proof
    let y be object;
    y in A implies y in x ++ A
    proof
      assume y in A;
      then reconsider w = y as Element of V;
      (w - x) + x = w - (x-x) by RLVECT_1:29
      .= w - 0.V by RLVECT_1:15
      .= w by RLVECT_1:13;
      hence y in x ++ A by P0;
    end;
    hence thesis by P0;
  end;
  hence x ++ A = A by TARSKI:2;
end;
