
theorem KERCL01:
  for X,Y be RealNormSpace,
      L be Function of X,Y,
      w be Point of Y
  st L is_continuous_on the carrier of X
  holds L"{w} is closed
  proof
    let X,Y be RealNormSpace,
        L be Function of X,Y,
        w be Point of Y;
    assume
    A1: L is_continuous_on the carrier of X;
    for seq be sequence of X st rng seq c= L"{w} & seq is convergent
    holds lim seq in L"{w}
    proof
      let seq be sequence of X;
      assume
      A2: rng seq c= L"{w} & seq is convergent;
      lim seq in the carrier of X; then
      A3: L /* seq is convergent & L /. (lim seq) = lim(L /* seq)
          by A1,A2,NFCONT_1:18;
      now
        let n be Nat;
        A4: n in NAT by ORDINAL1:def 12;
        seq.n in rng seq by FUNCT_2:4,ORDINAL1:def 12; then
        L.(seq.n) in {w} by A2,FUNCT_2:38; then
        L.(seq.n) = w by TARSKI:def 1;
        hence (L /* seq).n = w by A4,FUNCT_2:115;
      end; then
      lim(L /* seq) = w by CLOSE01; then
      L /. (lim seq) in {w} by A3,TARSKI:def 1;
      hence lim seq in L"{w} by FUNCT_2:38;
    end;
    hence thesis;
  end;
