reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;
reserve X for non empty set;
reserve p,p1,p2 for Point of TOP-REAL N;
reserve M for non empty MetrSpace;
reserve X,Y for non empty TopSpace;

theorem Th44:
  for f being Function of X,Y, P being non empty Subset of Y, f1
  being Function of X,Y|P st f=f1 & f is continuous holds f1 is continuous
proof
  let f be Function of X,Y, P be non empty Subset of Y, f1 be Function of X,Y|
  P;
  assume that
A1: f=f1 and
A2: f is continuous;
A3: [#]Y <> {};
A4: for P1 being Subset of (Y|P) st P1 is open holds f1"P1 is open
  proof
    let P1 be Subset of (Y|P);
    assume P1 is open;
    then P1 in the topology of (Y|P) by PRE_TOPC:def 2;
    then consider Q being Subset of Y such that
A5: Q in the topology of Y and
A6: P1 = Q /\ [#](Y|P) by PRE_TOPC:def 4;
    reconsider Q as Subset of Y;
A7: f"Q=f1"(rng f1 /\ Q) by A1,RELAT_1:133;
A8: [#](Y|P)=P by PRE_TOPC:def 5;
    then rng f1 /\ Q c= P /\ Q by XBOOLE_1:26;
    then
A9: f1"(rng f1 /\ Q) c= f1"P1 by A6,A8,RELAT_1:143;
    Q is open by A5,PRE_TOPC:def 2;
    then
A10: f"Q is open by A3,A2,TOPS_2:43;
    f1"P1 c= f"Q by A1,A6,RELAT_1:143,XBOOLE_1:17;
    hence thesis by A10,A7,A9,XBOOLE_0:def 10;
  end;
  [#](Y|P) <> {};
  hence thesis by A4,TOPS_2:43;
end;
