
theorem
for X,Y be RealNormSpace,T be non empty PartFunc of X,Y,
    S be non empty PartFunc of Y,X st
    T is closed & T is one-to-one & S=T"
    holds S is closed
proof
 let X,Y be RealNormSpace,T be non empty PartFunc of X,Y,
     S be non empty PartFunc of Y,X;
assume
  A1:T is closed & T is one-to-one & S=T";
  A2: rng T = dom S & dom T = rng S by A1, FUNCT_1:33;
for seq be sequence of Y
  st rng seq c= dom S & seq is convergent & S/*seq is convergent
  holds lim seq in dom S & lim (S/*seq)= S.(lim seq)
  proof
   let seq be sequence of Y;
   assume
A3: rng seq c= dom S & seq is convergent & S/*seq is convergent;
    set seq1=S/*seq;
A4:  rng seq1 c= dom T
    proof
      let x be object;
      assume x in rng seq1;
      then consider i be object such that
A5: i in dom seq1 & x=seq1.i by FUNCT_1:def 3;
      reconsider i as Nat by A5;
      S.(seq.i) in rng S by FUNCT_1:3, A3,NFCONT_1:5;
      hence x in dom T by A5, A2, A3,FUNCT_2:108;
    end;
A6:T/*seq1=seq
     proof
       now
         let n be Element of NAT;
         thus (T/*seq1).n = seq.n
         proof
     A7: seq.n in rng T by A3,NFCONT_1:5,A2;
           (T/*seq1).n= T.(seq1.n) by A4,FUNCT_2:108
                     .= T.(S.(seq.n)) by A3,FUNCT_2:108
                     .= seq.n by A1,A7,FUNCT_1:35;
           hence thesis;
         end;
       end;
       hence thesis;
     end;
     lim seq1 in dom T & lim (T/*seq1)=T.(lim S/*seq)
        by A1,A3,A4,A6,Th12;
     hence thesis by A2,A6,FUNCT_1:3,A1,FUNCT_1:34;
  end;
  hence thesis by Th12;
end;
