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 Th42:
  F ++ (G /\ H) c= (F++G) /\ (F++H)
proof
  let j;
  assume j in F++(G/\H);
  then consider w,w1 such that
A1: j = w+w1 and
A2: w in F and
A3: w1 in G/\H;
  w1 in H by A3,XBOOLE_0:def 4;
  then
A4: w+w1 in F++H by A2;
  w1 in G by A3,XBOOLE_0:def 4;
  then w+w1 in F++G by A2;
  hence thesis by A1,A4,XBOOLE_0:def 4;
end;
