
theorem Th1:
  for r being object holds r in REAL+ \ {0} iff r is positive Real
  proof
    let r be object;
    hereby
      assume r in REAL+ \ {0};
      then r in {r where r is Real: 0 <= r} & not r in {0}
        by REAL_1:1,XBOOLE_0:def 5;
      then r <> 0 & ex s be Real st r = s & 0 <= s by TARSKI:def 1;
      hence r is positive Real;
    end;
    assume r is positive Real;
    then r in {r where r is Real: 0 <= r} & not r in {0} by TARSKI:def 1;
    hence thesis by REAL_1:1,XBOOLE_0:def 5;
  end;
