
theorem Th3:
  for X,Y being non empty MetrSpace, f being Function of
TopSpaceMetr(X),TopSpaceMetr(Y),S being sequence of X, T being sequence of Y st
  S is convergent & T= f*S & f is continuous holds T is convergent
proof
  let X,Y be non empty MetrSpace, f be Function of TopSpaceMetr(X),
  TopSpaceMetr(Y),S be sequence of X,T be sequence of Y;
  assume that
A1: S is convergent and
A2: T= f*S and
A3: f is continuous;
  set z0=lim S;
  reconsider p=z0 as Point of TopSpaceMetr(X) by TOPMETR:12;
A4: dom f=the carrier of TopSpaceMetr(X) by FUNCT_2:def 1
    .=the carrier of X by TOPMETR:12;
  then f.(lim S) in rng f by FUNCT_1:def 3;
  then reconsider x0=f.(lim S) as Element of Y by TOPMETR:12;
  for r being Real st r>0
   ex n being Nat st for m being Nat st n<=m holds dist(T.m,x0)<r
  proof
    let r be Real;
    reconsider V=Ball(x0,r) as Subset of TopSpaceMetr(Y) by TOPMETR:12;
    assume r>0;
    then V is open & x0 in V by GOBOARD6:1,TOPMETR:14;
    then consider W being Subset of TopSpaceMetr(X) such that
A5: p in W & W is open and
A6: f.:W c= V by A3,JGRAPH_2:10;
    consider r0 being Real such that
A7: r0>0 and
A8: Ball(z0,r0) c= W by A5,TOPMETR:15;
    reconsider r00=r0 as Real;
    consider n0 being Nat such that
A9: for m being Nat st n0<=m holds dist(S.m,z0)<r00 by A1,A7,
TBSP_1:def 3;
    for m being Nat st n0<=m holds dist(T.m,x0)<r
    proof
      let m be Nat;
      assume n0<=m;
      then dist(S.m,z0)<r0 by A9;
      then S.m in Ball(z0,r0) by METRIC_1:11;
      then
A10:  f.(S.m) in f.:W by A4,A8,FUNCT_1:def 6;
A11:   m in NAT by ORDINAL1:def 12;
      dom T=NAT by FUNCT_2:def 1;
      then T.m in f.:W by A2,A10,FUNCT_1:12,A11;
      hence thesis by A6,METRIC_1:11;
    end;
    hence thesis;
  end;
  hence thesis by TBSP_1:def 2;
end;
