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

theorem
  for A being Subset of REAL st 0 in A & for x st x in A holds x + 1 in A
  holds NAT c= A
proof
  let A be Subset of REAL such that
A1: 0 in A and
A2: for x st x in A holds x + 1 in A;
  reconsider B = A /\ REAL+ as Subset of REAL+ by XBOOLE_1:17;
A3: B c= A by XBOOLE_1:17;
A4: for x9,y9 being Element of REAL+ st x9 in B & y9 = 1 holds x9 + y9 in B
  proof
    let x9,y9 be Element of REAL+ such that
A5: x9 in B and
A6: y9 = 1;
    reconsider x = x9 as Element of REAL by ARYTM_0:1;
    reconsider y = 1 as Element of REAL by NUMBERS:19;
    reconsider xx = x as Real;
    xx+1 in A by A2,A3,A5;
    then (ex x99,y99 being Element of REAL+ st x = x99 & 1 = y99 & +(x,y) =
    x99 + y99 )& +(x,y) in A by Lm3,ARYTM_0:def 1,ARYTM_2:20;
    hence thesis by A6,XBOOLE_0:def 4;
  end;
  0 in B by A1,ARYTM_2:20,XBOOLE_0:def 4;
  then NAT c= B by A4,ARYTM_2:17;
  hence thesis by A3;
end;
