reserve r,s for Real;
reserve x,y,z for Real;
reserve r,r1,r2 for Element of REAL+;

theorem
  x in NAT & y in NAT implies x + y in NAT
proof
  reconsider x1 = x, y1 = y as Element of REAL by XREAL_0:def 1;
A1: +(x1,y1) = x + y by Lm3;
  assume
A2: x in NAT & y in NAT;
  then ex x9,y9 being Element of REAL+ st x1 = x9 & y1 = y9 & +(x1,y1) = x9 +
  y9 by ARYTM_0:def 1,ARYTM_2:2;
  hence thesis by A1,A2,ARYTM_2:16;
end;
