 reserve x for Real,
    p,k,l,m,n,s,h,i,j,k1,t,t1 for Nat,
    X for Subset of REAL;

theorem Th1:
  for X being Subset of REAL st
    0 in X &
    for x being Real st x in X holds x + 1 in X holds
  for n being Nat holds n in X
proof
  let A be Subset of REAL such that
A1: 0 in A;
  assume x in A implies x + 1 in A;
  then NAT c= A by A1,AXIOMS:3;
  hence thesis by ORDINAL1:def 12;
end;
