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
  s < r + t & t <> {} implies ex r0,t0 being Element of RAT+ st s = r0 +
  t0 & r0 <=' r & t0 <=' t & t0 <> t
proof
  assume that
A1: s < r + t and
A2: t <> {};
  per cases;
  suppose
    r <=' s;
    then consider t0 being Element of RAT+ such that
A3: s = r + t0 & t0 <=' t by A1,Th86;
    take r,t0;
    thus thesis by A1,A3;
  end;
  suppose
A4: s <=' r;
    s = s+{} & {} <=' t by Th50,Th64;
    hence thesis by A2,A4;
  end;
end;
