theorem Th32:
  for k,n being natural Number holds
    k in Segm n iff k < n
proof
  let k,n be natural Number;
  hereby
    assume k in Segm n;
    then k in { l where l is Nat : l < n } by AXIOMS:4;
    then ex l being Nat st k = l & l < n;
    hence k < n;
  end;
  assume
A1: k < n;
  reconsider k as Element of NAT by ORDINAL1:def 12;
  k in { l where l is Nat : l < n } by A1;
  hence thesis by AXIOMS:4;
end;
