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 Th40:
  f is_continuous_on X iff f|X is_continuous_on X
proof
  thus f is_continuous_on X implies f|X is_continuous_on X
  proof
    assume
A1: f is_continuous_on X;
    then X c= dom f;
    then X c= dom f /\ X by XBOOLE_1:28;
    hence X c= dom (f|X) by RELAT_1:61;
    let g;
    assume g in X;
    then f|X is_continuous_in g by A1;
    hence thesis by RELAT_1:72;
  end;
  assume
A2: f|X is_continuous_on X;
  then X c= dom(f|X);
  then dom f /\ X c= dom f & X c= dom f /\ X by RELAT_1:61,XBOOLE_1:17;
  hence X c= dom f;
  let g;
  assume g in X;
  then (f|X)|X is_continuous_in g by A2;
  hence thesis by RELAT_1:72;
end;
