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
  f1 is_continuous_on X & f1"{0} = {} & f2 is_continuous_on X implies f2
  /f1 is_continuous_on X
proof
  assume that
A1: f1 is_continuous_on X & f1"{0} = {} and
A2: f2 is_continuous_on X;
  f1^ is_continuous_on X by A1,Th47;
  then f2(#)(f1^) is_continuous_on X by A2,Th43;
  hence thesis by CFUNCT_1:39;
end;
