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+;

theorem Th39:
  (numerator x)/(denominator x) = x
proof
A1: denominator x <> {} by Th35;
A2: RED(numerator x, denominator x) = numerator x by Th23,Th34;
  numerator x, denominator x are_coprime by Th34;
  then
A3: RED(denominator x, numerator x) = denominator x by Th23;
  per cases;
  suppose
A4: denominator x = 1;
    then x in omega by Th36;
    then numerator x = x by Def8;
    hence thesis by A2,A3,A4,Def10;
  end;
  suppose
A5: denominator x <> 1;
    then not x in omega by Def9;
    then x = [numerator x, denominator x] by Th36;
    hence thesis by A1,A2,A3,A5,Def10;
  end;
end;
