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 Th48:
  A ++ (B \/ C) = (A++B) \/ (A++C)
proof
  let a;
  hereby
    assume a in A++(B\/C);
    then consider c,c1 such that
A1: a = c+c1 and
A2: c in A and
A3: c1 in B\/C;
    c1 in B or c1 in C by A3,XBOOLE_0:def 3;
    then c+c1 in A++B or c+c1 in A++C by A2;
    hence a in (A++B)\/(A++C) by A1,XBOOLE_0:def 3;
  end;
  assume
A4: a in (A++B)\/(A++C);
  per cases by A4,XBOOLE_0:def 3;
  suppose
    a in A++B;
    then consider c,c1 such that
A5: a = c+c1 & c in A and
A6: c1 in B;
    c1 in B\/C by A6,XBOOLE_0:def 3;
    hence thesis by A5;
  end;
  suppose
    a in A++C;
    then consider c,c1 such that
A7: a = c+c1 & c in A and
A8: c1 in C;
    c1 in B\/C by A8,XBOOLE_0:def 3;
    hence thesis by A7;
  end;
end;
