reserve s,t for Element of RAT+;

theorem
  REAL+ = { r where r is Real: 0 <= r}
 proof set RP = { r where r is Real: 0 <= r};
  thus REAL+ c= RP
   proof let e be object;
    assume
A1:     e in REAL+;
      then reconsider r = e as Real by ARYTM_0:1;
      reconsider o = 0, s = r as Element of REAL+ by A1,ARYTM_2:20;
      o <=' s by ARYTM_1:6;
      then 0 <= r by ARYTM_2:20,XXREAL_0:def 5;
      hence e in RP;
   end;
  let e be object;
  assume e in RP;
   then
A2:   ex r being Real st e = r & 0 <= r;
   not 0 in [:{0},REAL+:] by ARYTM_0:5,ARYTM_2:20,XBOOLE_0:3;
  hence e in REAL+ by A2,XXREAL_0:def 5;
 end;
