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 Th190:
  (f**F) \ (f**G) c= f ** (F \ G)
proof
  let i;
  assume
A1: i in (f**F) \ (f**G);
  then consider w such that
A2: i = f*w and
A3: w in F by Th188;
  now
    assume not w in F\G;
    then w in G by A3,XBOOLE_0:def 5;
    then f*w in f**G by Th186;
    hence contradiction by A1,A2,XBOOLE_0:def 5;
  end;
  hence thesis by A2,Th186;
end;
