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 Th82:
  F ** (G \/ H) = (F**G) \/ (F**H)
proof
  let j;
  hereby
    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 G or w1 in H by A3,XBOOLE_0:def 3;
    then w*w1 in F**G or w*w1 in F**H by A2;
    hence j in (F**G)\/(F**H) by A1,XBOOLE_0:def 3;
  end;
  assume
A4: j in (F**G)\/(F**H);
  per cases by A4,XBOOLE_0:def 3;
  suppose
    j in F**G;
    then consider w,w1 such that
A5: j = w*w1 & w in F and
A6: w1 in G;
    w1 in G\/H by A6,XBOOLE_0:def 3;
    hence thesis by A5;
  end;
  suppose
    j in F**H;
    then consider w,w1 such that
A7: j = w*w1 & w in F and
A8: w1 in H;
    w1 in G\/H by A8,XBOOLE_0:def 3;
    hence thesis by A7;
  end;
end;
