reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem Th3:
  for TX,TY being non empty TopSpace,
  P being non empty Subset of TY,f being Function of TX,TY|P
  holds f is Function of TX,TY & for f2 being Function of TX,TY
  st f2=f & f is continuous holds f2 is continuous
proof
  let TX,TY be non empty TopSpace,
  P be non empty Subset of TY,f be Function of TX,TY|P;
A1: the carrier of TY|P = [#](TY|P) .=P by PRE_TOPC:def 5;
  hence f is Function of TX,TY by FUNCT_2:7;
  let f2 be Function of TX,TY such that
A2: f2=f and
A3: f is continuous;
  let P1 be Subset of TY;
  assume
A4: P1 is closed;
  reconsider P3=P1/\P as Subset of TY|P by TOPS_2:29;
A5: P3 is closed by A4,Th2;
  f2"P=the carrier of TX by A1,A2,FUNCT_2:40
    .=dom f2 by FUNCT_2:def 1;
  then f2"P1=f2"P1 /\ f2"P by RELAT_1:132,XBOOLE_1:28
    .=f"P3 by A2,FUNCT_1:68;
  hence thesis by A3,A5;
end;
