
theorem Th12:
  for X,Y be RealNormSpace, T be non empty PartFunc of X,Y holds
  T is closed iff
  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 X,Y be RealNormSpace, T be non empty PartFunc of X,Y;
hereby
assume A1: T is closed;
thus 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 A2: rng seq c= dom T & seq is convergent & T/*seq is convergent;
 set s1 = <:seq,T/*seq:>;
  A3: rng s1 c= graph(T)
    proof
     let y be object;
     assume y in rng s1;
     then consider i be object
     such that A4: i in NAT & s1.i = y by FUNCT_2:11;
A5: (T/*seq).i = T.(seq.i) by A4,A2,FUNCT_2:108;
     seq.i in rng seq by A4,FUNCT_2:4;
     then [seq.i,(T/*seq).i ] in T by A5,FUNCT_1:def 2,A2;
     hence y in graph(T) by A4,FUNCT_3:59;
   end;
   lim seq = lim seq & lim (T/*seq) = lim (T/*seq);
    then s1 is convergent
     & lim s1 = [lim seq, lim (T/*seq) ] by Th8,A2;
      then [lim seq, lim (T/*seq) ] in graph(T)
             by A1,NFCONT_1:def 3,A3;
      hence lim seq in dom T & lim (T/*seq)= T.(lim seq) by FUNCT_1:1;
 end;
end;
assume
A6: 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);
for s1 be sequence of [:X,Y:] st rng s1 c= graph(T)
          & s1 is convergent holds lim s1 in graph(T)
proof
 let s1 be sequence of [:X,Y:];
  assume A7: rng s1 c= graph(T) & s1 is convergent;
 defpred Q0[set,set] means [$2,T.$2] = s1.$1;
 A8: for i being Element of NAT ex x being Element of
        the carrier of X st Q0[i,x]
 proof
  let i be Element of NAT;
A9: s1.i in rng s1 by FUNCT_2:4;
   consider x be Point of X,y be Point of Y such that
   A10: s1.i =[x,y] by PRVECT_3:18;
   take x;
   thus thesis by A10,FUNCT_1:1,A9,A7;
 end;
 consider seq be sequence of X such that
   A11: for x being Element of NAT holds Q0[x,seq.x] from FUNCT_2:sch 3(A8);
A12:  now let y be object;
        assume y in rng seq; then
        consider i be object such that
A13:    i in dom seq & y=seq. i by FUNCT_1:def 3;
    A14: [seq.i,T.(seq.i) ] = s1.i by A13,A11;
        s1.i in rng s1 by A13,FUNCT_2:4;
        hence y in dom T by A13,FUNCT_1:1, A14,A7;
      end; then
   A15:rng seq c= dom T;
   consider x be Point of X,y be Point of Y such that
   A16: lim s1 =[x,y] by PRVECT_3:18;
   s1 = <:seq,T/*seq:>
   proof
     let n be Element of NAT;
     (T/*seq).n = T.(seq.n) by A12,TARSKI:def 3,FUNCT_2:108;
     hence s1.n = [seq.n,(T/*seq).n] by A11
     .= <:seq,T/*seq:>.n by FUNCT_3:59;
   end; then
   A17: seq is convergent & lim seq = x
       & T/*seq is convergent & lim (T/*seq) = y by A16,Th8,A7;
    lim seq in dom T & lim (T/*seq)= T.(lim seq) by A15,A6,A17;
    hence lim s1 in graph(T) by A16,A17,FUNCT_1:1;
  end;
  hence thesis by NFCONT_1:def 3;
end;
