reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th2:
  NATPLUS = NAT \ {0}
  proof
A1: NATPLUS c= NAT \ {0}
    proof
      let x be object;
      assume
A2:   x in NATPLUS;
      then reconsider x as Element of NAT;
      not x in {0} by A2,TARSKI:def 1;
      hence thesis by XBOOLE_0:def 5;
    end;
    NAT \ {0} c= NATPLUS
    proof
      let x be object;
      assume
A3:   x in NAT \ {0};
      then reconsider x as Element of NAT;
      not x in {0} by A3,XBOOLE_0:def 5;
      then x <> 0 by TARSKI:def 1;
      hence thesis by NAT_LAT:def 6;
    end;
    hence thesis by A1,XBOOLE_0:def 10;
  end;
