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 Th71:
  x in omega & x+y in omega implies y in omega
proof
  assume that
A1: x in omega and
A2: x+y in omega;
A3: denominator (x+y) = 1 by A2,Def9;
  set nx = numerator x, dx = denominator x;
A4: dx = 1 by A1,Def9;
  set ny = numerator y, dy = denominator y;
A5: x+y = numerator (x+y)/denominator (x+y) & (nx*^dy+^ny*^dx)*^1 = nx*^dy+^
  ny*^ dx by Th39,ORDINAL2:39;
  dy <> {} by Th35;
  then dx*^dy <> {} by A4,ORDINAL3:31,Lm4;
  then nx*^dy+^ny*^dx = dx*^dy*^numerator (x+y) by A3,A5,Th45,Lm4
    .= dy*^numerator (x+y) by A4,ORDINAL2:39;
  then nx*^dy+^ny = dy*^numerator (x+y) by A4,ORDINAL2:39;
  then ny = (dy*^numerator (x+y))-^nx*^dy by ORDINAL3:52;
  then ny = dy*^((numerator (x+y))-^nx) by ORDINAL3:63;
  then
A6: dy divides ny;
A7: ny,dy are_coprime by Th34;
  for m being natural Ordinal st m divides dy & m divides ny holds m
  divides dy;
  then dy hcf ny = dy by A6,Def5;
  then
A8: dy = 1 by A7,Th20;
  y = ny/dy by Th39;
  then y = ny by A8,Th40;
  hence thesis;
end;
