
theorem Th4:
  for n being Ordinal, T being admissible TermOrder of n, b1,b2,b3
  being bag of n holds b1 < b2,T implies b1 + b3 < b2 + b3,T
proof
  let n be Ordinal, T be admissible TermOrder of n, b1,b2,b3 be bag of n;
  assume
A1: b1 < b2,T;
A2: now
    assume
A3: b1+b3 = b2+b3;
    b1 = (b1 + b3) -' b3 by PRE_POLY:48
      .= b2 by A3,PRE_POLY:48;
    hence contradiction by A1,TERMORD:def 3;
  end;
  b1 <= b2,T by A1,TERMORD:def 3;
  then [b1,b2] in T by TERMORD:def 2;
  then [b1+b3,b2+b3] in T by BAGORDER:def 5;
  then b1+b3 <= b2+b3,T by TERMORD:def 2;
  hence thesis by A2,TERMORD:def 3;
end;
