theorem Th22:
for k being Nat holds k+1 in SUCC(k,SCM) &
 for j being Nat st j in SUCC(k,SCM) holds k <= j
proof
  let k be Nat;
  reconsider fk = k as Element of NAT by ORDINAL1:def 12;
A1: SUCC(k,SCM) = {k, fk+1} by Th21;
  hence k+1 in SUCC(k,SCM) by TARSKI:def 2;
  let j be Nat;
  assume
A2: j in SUCC(k,SCM);
  reconsider fk = k as Element of NAT by ORDINAL1:def 12;
  per cases by A1,A2,TARSKI:def 2;
  suppose
    j = k;
    hence thesis;
  end;
  suppose
    j = fk + 1;
    hence thesis by NAT_1:11;
  end;
end;
