
theorem Th101:
  for a, b, c being Ordinal st a in exp(omega,c) & b in exp(omega,c)
  holds a (+) b in exp(omega,c)
proof
  let a, b, c be Ordinal;
  assume A1: a in exp(omega,c) & b in exp(omega,c);
  per cases;
  suppose a = 0;
    hence thesis by A1, Th82;
  end;
  suppose b = 0;
    hence thesis by A1, Th82;
  end;
  suppose A2: a <> {} & b <> {};
    then 0 c< a & 0 c< b by XBOOLE_1:2, XBOOLE_0:def 8;
    then 0 in a & 0 in b by ORDINAL1:11;
    then omega -exponent a in c & omega -exponent b in c by A1, Th23;
    then omega -exponent a \/ omega -exponent b in c by ORDINAL3:12;
    then A3: omega -exponent(a(+)b) in c by Th100;
    A4: not c c= omega -exponent(a(+)b)
    proof
      assume c c= omega -exponent(a(+)b);
      then omega-exponent(a(+)b) in omega-exponent(a(+)b) by A3;
      hence contradiction;
    end;
    0 c< a (+) b by A2, XBOOLE_1:2, XBOOLE_0:def 8;
    then 0 in a (+) b by ORDINAL1:11;
    then not exp(omega,c) c= a (+) b by A4, ORDINAL5:def 10;
    hence thesis by ORDINAL1:16;
  end;
end;
