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 Th63:
  r+s = {} implies r = {}
proof
  set nr = numerator r, dr = denominator r;
  set ns = numerator s, ds = denominator s;
  assume
A1: r+s = {};
  denominator {} = 1 & numerator {} = {} by Def8,Def9,Lm1;
  then
A2: (nr*^ds+^ns*^dr)/(dr*^ds) = {}/1 by A1,Th39;
A3: ds <> {} by Th35;
  dr <> {} by Th35;
  then dr*^ds <> {} by A3,ORDINAL3:31;
  then (nr*^ds+^ns*^dr)*^1 = (dr*^ds)*^{} by A2,Th45,Lm4
    .= {} by ORDINAL2:35;
  then nr*^ds+^ns*^dr = {} by ORDINAL3:31,Lm4;
  then nr*^ds = {} by ORDINAL3:26;
  then nr = {} by A3,ORDINAL3:31;
  hence thesis by Th37;
end;
