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
  t <> {} implies ex s st s in omega & r <=' s *' t
proof
  set y=((numerator r)*^(((numerator t)*^denominator r)-^1))/denominator r;
A1: denominator r <> {} by Th35;
  (denominator t)*^(numerator r) in omega by ORDINAL1:def 12;
  then reconsider
  s = (denominator t)*^(numerator r) as ordinal Element of RAT+ by Lm6;
  assume t <> {};
  then numerator t <> {} by Th37;
  then {} in (numerator t)*^denominator r by A1,ORDINAL3:8,31;
  then one c= (numerator t)*^denominator r by Lm3,ORDINAL1:21;
  then (numerator t)*^denominator r = 1+^(((numerator t)*^denominator r)-^ 1)
  by ORDINAL3:def 5;
  then
  s*^((numerator t)*^denominator r) = (denominator t)*^ ((numerator r)*^(1
  +^(((numerator t)*^denominator r)-^1))) by ORDINAL3:50
    .= (denominator t)*^((numerator r)*^1+^ (numerator r)*^(((numerator t)*^
  denominator r)-^1)) by ORDINAL3:46;
  then
A2: s*^(numerator t)*^denominator r = (denominator t)*^((numerator r)*^1+^
  (numerator r)*^(((numerator t)*^denominator r)-^1)) by ORDINAL3:50;
  take s;
  thus s in omega by ORDINAL1:def 12;
  take y;
  denominator t <> {} by Th35;
  then
  (s*^(numerator t))/denominator t = ((numerator r)*^1+^ (numerator r)*^(
  ((numerator t)*^denominator r)-^1))/denominator r by A1,A2,Th45
    .= (((numerator r)*^1)/denominator r)+y by Th35,Th47
    .= ((numerator r)/denominator r)+y by ORDINAL2:39
    .= r+y by Th39;
  then r+y = (s*^(numerator t))/(1*^denominator t) by ORDINAL2:39
    .= (s/1)*'((numerator t)/denominator t) by Th49
    .= s*'((numerator t)/denominator t) by Th40;
  hence thesis by Th39;
end;
