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
  f is_continuous_on X & (f|X)"{0} = {} implies f^ is_continuous_on X
proof
  assume that
A1: f is_continuous_on X and
A2: (f|X)"{0} = {};
  f|X is_continuous_on X by A1,Th40;
  then (f|X)^ is_continuous_on X by A2,Th47;
  then (f^)|X is_continuous_on X by CFUNCT_1:54;
  hence thesis by Th40;
end;
