reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;
reserve i,j,k for Element of omega;
reserve x,y,z for Element of RAT+;
reserve i,j,k for natural Ordinal;
reserve r,s,t for Element of RAT+;

theorem
  for r1,r2,s1,s2 being Element of RAT+ st r1*'r2 = s1*'s2 holds r1 <='
  s1 or r2 <=' s2
proof
  let r1,r2,s1,s2 be Element of RAT+ such that
A1: r1*'r2 = s1*'s2;
A2: {} <=' s1 by Th64;
  assume that
A3: s1 < r1 and
A4: s2 < r2;
  s2 <> r2 & s1*'s2 <=' r1*'s2 by A3,A4,Th82;
  then s1*'s2 < r1*'s2 by A1,A3,A2,Th56,Th66;
  hence thesis by A1,A4,Th82;
end;
