reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;
reserve i for Nat;
reserve m for Nat,
        D for non empty set;

theorem
  for F being initial NAT-defined finite Function
  holds dom F = { k where k is Element of NAT: k < card F }
proof
  let F be initial NAT-defined finite Function;
  hereby
    let x be object;
    assume
A1: x in dom F;
    then reconsider f = x as Element of NAT;
    f < card F by A1,Lm1;
    hence x in { k where k is Element of NAT: k < card F };
  end;
  let x be object;
  assume x in { k where k is Element of NAT: k < card F };
  then ex k being Element of NAT st x = k & k < card F;
  hence thesis by Lm1;
end;
