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 Th73:
  t <> {} implies ex r st r < t & not r in omega
proof
  assume
A1: t <> {};
A2: 1+^1 = succ 1 by ORDINAL2:31;
  per cases;
  suppose
A3: one <=' t;
    consider r such that
A4: one = r+r by Th60;
    take r;
    r <=' one & r <> 1 by A2,A4,Th58;
    then r < one by Th66;
    hence r < t by A3,Th67;
A5: 1*^1 = 1 by ORDINAL2:39;
    assume r in omega;
    then
A6: denominator r = 1 by Def9;
    then (numerator r)*^denominator r = numerator r by ORDINAL2:39;
    then
A7: 1 = (numerator r)+^numerator r by A4,A6,A5,Th40;
    then numerator r c= 1 by ORDINAL3:24;
    then numerator r = {} or numerator r = 1 by ORDINAL3:16;
    hence contradiction by A2,A7,ORDINAL2:27;
  end;
  suppose
A8: t < one;
    consider r such that
A9: t = r+r by Th60;
A10: {} <=' r by Th64;
    r <> {} by A1,A9,Th50;
    then
A11: {} < r by A10,Th66;
    take r;
A12: 1 = {}+one by Th50;
A13: r <=' t by A9;
    now
      assume r = t;
      then t+{} = t+t by A9,Th50;
      hence contradiction by A1,Th62;
    end;
    hence r < t by A13,Th66;
    r < one by A8,A13,Th67;
    hence thesis by A11,A12,Th72;
  end;
end;
