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: s < t} in RAT+ iff t = {}
proof
  hereby
    assume
A1: {s: s < t} in RAT+ & t <> {};
    then consider r such that
A2: r < t and
A3: not r in omega by Th73;
    {} <=' t by Th64;
    then {} < t by A1,Th66;
    then
A4: {} in {s: s < t};
    r in {s: s < t} by A2;
    then {s: s < t} in omega implies r is Ordinal;
    then ex i,j being Element of omega st {s: s < t} = [i,j] & i,j
    are_coprime & j <> {} & j <> 1 by A1,A3,Th29,Th31;
    hence contradiction by A4,TARSKI:def 2;
  end;
  assume
A5: t = {};
  {s: s < t} c= {}
  proof
    let a be object;
    assume a in {s: s < t};
    then ex s st a = s & s < t;
    hence thesis by A5,Th64;
  end;
  then {s: s < t} = {};
  hence thesis;
end;
