reserve r,s,t,x9,y9,z9,p,q for Element of RAT+;
reserve x,y,z for Element of REAL+;

theorem
  for x st x in omega holds for y holds y in x iff y in omega & y <> x &
  y <=' x
proof
  let x;
  assume
A1: x in omega;
  then reconsider m = x as Element of omega;
  reconsider x9 = x as Element of RAT+ by A1,Lm5;
  let y;
A2: x c= omega by A1,ORDINAL1:def 2;
  hereby
    assume
A3: y in x;
    then reconsider n = y as Element of omega by A2;
    thus y in omega by A2,A3;
    then reconsider y9 = y as Element of RAT+ by Lm5;
    thus y <> x by A3;
    n c= m by A3,ORDINAL1:def 2;
    then consider C being Ordinal such that
A4: m = n +^ C by ORDINAL3:27;
    C c= m by A4,ORDINAL3:24;
    then reconsider C as Element of omega by ORDINAL1:12;
    reconsider z9 = C as Element of RAT+ by Lm5;
    x9 = y9 + z9 by A4,Lm45;
    then y9 <=' x9;
    hence y <=' x by Lm14;
  end;
  assume
A5: y in omega;
  then reconsider y9 = y as Element of RAT+ by Lm5;
  reconsider n = y as Element of omega by A5;
  assume
A6: y <> x;
  assume y <=' x;
  then y9 <=' x9 by Lm14;
  then consider z9 such that
A7: y9 + z9 = x9;
  reconsider k = z9 as Element of omega by A1,A5,A7,ARYTM_3:71;
  n +^ k = m by A7,Lm45;
  then n c= m by ORDINAL3:24;
  then n c< m by A6;
  hence thesis by ORDINAL1:11;
end;
