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

theorem
  for k being natural Number holds k = { i where i is Nat: i < k }
proof
  let k be natural Number;
A1: k in NAT by ORDINAL1:def 12;
  set Y = { i where i is Nat: i < k };
  reconsider K = k as Element of NAT by ORDINAL1:def 12;
  for e being object holds e in K iff e in Y
  proof
    let e be object;
    thus e in K implies e in Y
    proof
      assume
A2:   e in K;
A3:   K c= NAT by ORDINAL1:def 2;
      then reconsider j = e as Element of NAT by A2;
      e in NAT by A3,A2;
      then reconsider y9 = e as Element of REAL+ by ARYTM_2:2;
      reconsider x9 = K as Element of REAL+ by ARYTM_2:2;
      y9 <=' x9 by A2,ARYTM_2:18;
      then
A4:   j <= k by Lm1;
      reconsider yy9 = y9 as set;
      not yy9 in yy9; then
      y9 <> x9 by A2;
      then j < k by A4,XXREAL_0:1;
      hence thesis;
    end;
    assume e in Y;
    then consider i be Nat such that
A5: e = i and
A6: not k <= i;
A7: i in NAT by ORDINAL1:def 12;
    then reconsider x9 = e, y9 = k as Element of REAL+ by A5,A1,ARYTM_2:2;
    not y9 <=' x9 by A5,A6,Lm1;
    hence thesis by A5,A6,A7,ARYTM_2:18;
  end;
  hence thesis by TARSKI:2;
end;
