reserve r,s,t,x9,y9,z9,p,q for Element of RAT+;
reserve x,y,z for Element of REAL+;

theorem
  x in omega & y in omega implies y + x in omega
proof
  assume
A1: x in omega & y in omega;
  then reconsider x0 = x, y0 = y as Element of omega;
  consider x9,y9 being Element of RAT+ such that
A2: x = x9 & y = y9 and
A3: x + y = x9 + y9 by A1,Lm5,Lm47;
  x9 + y9 = x0 +^ y0 by A2,Lm45;
  hence thesis by A3,ORDINAL1:def 12;
end;
