
theorem
  for T being Nat st T>0 holds
    {w where w is Element of NAT: w>0 & w <= T} is non empty Subset of NAT
proof
 let T be Nat;
 assume A0: T>0;
  B1: {w where w is Element of NAT: w>0 & w <= T} c= NAT
  proof
    let x be object;
    assume x in {w where w is Element of NAT: w>0 & w <= T};
    then consider w being Element of NAT such that C1: x=w & w>0 & w <= T;
   thus thesis by C1;
  end;
  1>0 & 1<=T
  proof
    T>0 & T=1*T by A0;
    hence thesis by NAT_1:24;
  end; then
  1 in {w where w is Element of NAT: w>0 & w <= T};
  hence thesis by B1;
end;
