
theorem
  for X,Y be RealNormSpace,
      T be non empty PartFunc of X,Y,T0 be LinearOperator of X,Y st
   T0 is Lipschitzian & dom T is closed & T=T0
    holds T is closed
proof
  let X,Y be RealNormSpace,
      T be non empty PartFunc of X,Y,T0 be LinearOperator of X,Y;
  assume
A1:T0 is Lipschitzian & dom T is closed & T=T0; then
A2:T is_continuous_in 0.X by Th5,Th6;
  for seq be sequence of X
  st rng seq c= dom T & seq is convergent & T/*seq is convergent
  holds lim seq in dom T & lim (T/*seq)= T.(lim seq)
 proof
   let seq be sequence of X;
   assume
A3: rng seq c= dom T & seq is convergent & T/*seq is convergent;
 A4:T is_continuous_in lim seq by A1,A2,Th4;
   T/.(lim seq)=T.(lim seq) by A1,A3,NFCONT_1:def 3,PARTFUN1:def 6;
   hence thesis by A3,A4, NFCONT_1:def 5;
 end;
 hence thesis by Th12;
end;
