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 non empty Subset of RAT+ st A in RAT+ ex s st s in A & for
  r st r in A holds r <=' s
proof
  let A be non empty Subset of RAT+;
A1: now
    given i,j being Element of omega such that
A2: A = [i,j] and
A3: i,j are_coprime and
    j <> {} and
A4: j <> 1;
A5: now
      assume {i} in omega;
      then {} in {i} by ORDINAL3:8;
      then {} = i by TARSKI:def 1;
      hence contradiction by A3,A4,Th3;
    end;
    {i} in A by A2,TARSKI:def 2;
    then consider i1,j1 being Element of omega such that
A6: {i} = [i1,j1] and
A7: i1,j1 are_coprime and
    j1 <> {} and
A8: j1 <> 1 by A5,Th29;
    {i1,j1} in {i} by A6,TARSKI:def 2;
    then
A9: i = {i1,j1} by TARSKI:def 1;
    {i1} in {i} by A6,TARSKI:def 2;
    then i = {i1} by TARSKI:def 1;
    then j1 in {i1} by A9,TARSKI:def 2;
    then
A10: j1 = i1 by TARSKI:def 1;
    j1 = j1*^1 by ORDINAL2:39;
    hence contradiction by A7,A8,A10;
  end;
  assume A in RAT+;
  then reconsider B = A as Element of omega by A1,Th29;
A11: {} in B by ORDINAL3:8;
  now
    assume B is limit_ordinal;
    then omega c= B by A11,ORDINAL1:def 11;
    hence contradiction by ORDINAL1:5;
  end;
  then consider C such that
A12: B = succ C by ORDINAL1:29;
  C in B by A12,ORDINAL1:6;
  then reconsider C as ordinal Element of RAT+;
  take C;
  thus C in A by A12,ORDINAL1:6;
  let r;
  assume
A13: r in A;
  then r in B;
  then reconsider r as ordinal Element of RAT+;
  C-^r in omega by ORDINAL1:def 12;
  then reconsider x = C-^r as ordinal Element of RAT+ by Lm6;
  r c= C by A12,A13,ORDINAL1:22;
  then C = r+^x by ORDINAL3:def 5
    .= r+x by Th58;
  hence thesis;
end;
