reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem Th34:
  f is_continuous_in x0 implies g(#)f is_continuous_in x0
proof
  assume
A1: f is_continuous_in x0;
  then x0 in dom f;
  hence
A2: x0 in dom (g(#)f) by VALUED_1:def 5;
  let s1;
  assume that
A3: rng s1 c= dom(g(#)f) and
A4: s1 is convergent & lim s1=x0;
A5: rng s1 c= dom f by A3,VALUED_1:def 5;
  then
A6: f/.x0 = lim (f/*s1) by A1,A4;
A7: f/*s1 is convergent by A1,A4,A5;
  then g(#)(f/*s1) is convergent;
  hence (g(#)f)/*s1 is convergent by A5,Th8;
  thus (g(#)f)/.x0 =g*(f/.x0) by A2,CFUNCT_1:4
    .= lim (g(#)(f/*s1)) by A7,A6,COMSEQ_2:18
    .= lim ((g(#)f)/*s1) by A5,Th8;
end;
