reserve w, w1, w2 for Element of ExtREAL;
reserve c, c1, c2 for Complex;
reserve A, B, C, D for complex-membered set;
reserve F, G, H, I for ext-real-membered set;
reserve a, b, s, t, z for Complex;
reserve f, g, h, i, j for ExtReal;
reserve r for Real;
reserve e for set;

theorem Th191:
  r <> 0 implies r ** (F\G) = (r**F) \ (r**G)
proof
  assume
A1: r <> 0;
A2: r ** (F \ G) c= (r**F) \ (r**G)
  proof
    let i;
    assume i in r ** (F \ G);
    then consider w such that
A3: i = r*w and
A4: w in F\G by Th188;
A5: now
      assume r*w in r**G;
      then consider w1 such that
A6:   r*w = r*w1 and
A7:   w1 in G by Th188;
      w = w1 by A1,A6,XXREAL_3:68;
      hence contradiction by A4,A7,XBOOLE_0:def 5;
    end;
    r*w in r**F by A4,Th186;
    hence thesis by A3,A5,XBOOLE_0:def 5;
  end;
  (r**F) \ (r**G) c= r ** (F \ G) by Th190;
  hence thesis by A2;
end;
