reserve r,s,t,u for Real;

theorem Th15:
  for X being RealLinearSpace, M,N being Subset of X, r being non
  zero Real holds (r * M) /\ (r * N) = r * (M /\ N)
proof
  let X be RealLinearSpace, M,N be Subset of X, r be non zero Real;
  thus for x being object st x in (r * M) /\ (r * N)
   holds x in r * (M /\ N)
  proof
    let x be object;
    assume
A1: x in (r * M) /\ (r * N);
    then x in r * M by XBOOLE_0:def 4;
    then consider v1 being Point of X such that
A2: r*v1 = x and
A3: v1 in M;
    x in (r * N) by A1,XBOOLE_0:def 4;
    then consider v2 being Point of X such that
A4: r*v2 = x and
A5: v2 in N;
    v1 = v2 by A2,A4,RLVECT_1:36;
    then v1 in M /\ N by A3,A5,XBOOLE_0:def 4;
    hence thesis by A2;
  end;
  let x be object;
  assume x in r * (M /\ N);
  then consider v being Point of X such that
A6: r*v = x and
A7: v in M /\ N;
  v in N by A7,XBOOLE_0:def 4;
  then
A8: x in r*N by A6;
  v in M by A7,XBOOLE_0:def 4;
  then x in r*M by A6;
  hence thesis by A8,XBOOLE_0:def 4;
end;
