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 A being Subset of RAT+ st (ex t st t in A & t <> {}) & for r,s st
r in A & s <=' r holds s in A ex r1,r2,r3 being Element of RAT+ st r1 in A & r2
  in A & r3 in A & r1 <> r2 & r2 <> r3 & r3 <> r1
proof
  let A be Subset of RAT+;
  given t such that
A1: t in A and
A2: t <> {};
  assume
A3: for r,s st r in A & s <=' r holds s in A;
  consider x such that
A4: t = x+x by Th60;
  take {}, x, t;
  x <=' t by A4;
  hence {} in A & x in A & t in A by A1,A3,Th64;
  thus {} <> x by A2,A4,Th50;
  hereby
    assume x = t;
    then t+{} = t+t by A4,Th50;
    hence contradiction by A2,Th62;
  end;
  thus thesis by A2;
end;
