
theorem Th8:
  for T,S being non empty TopSpace holds for f being Function of T
,S holds for P being Subset of T holds P is compact & f is continuous implies f
  .:P is compact
proof
  let T,S be non empty TopSpace;
  let f be Function of T,S;
  let P be Subset of T;
  assume that
A1: P is compact and
A2: f is continuous;
  P c= [#]T;
  then
A3: P c= dom f by FUNCT_2:def 1;
  for F0 being Subset-Family of S st F0 is Cover of f.:P & F0 is open ex G
  being Subset-Family of S st G c= F0 & G is Cover of f.:P & G is finite
  proof
    let F0 be Subset-Family of S;
    assume that
A4: F0 is Cover of f.:P and
A5: F0 is open;
    set B0 = f"F0;
A6: f.:P c= union F0 by A4,SETFAM_1:def 11;
    P c= union B0
    proof
      let x be object;
      thus x in P implies x in union B0
      proof
A7:     f"(union F0) c= union(f"F0)
        proof
          let y be object;
          assume
A8:       y in f"(union F0);
          thus y in union(f"F0)
          proof
            f.y in union F0 by A8,FUNCT_1:def 7;
            then consider Q being set such that
A9:         f.y in Q & Q in F0 by TARSKI:def 4;
A10:        y in dom f by A8,FUNCT_1:def 7;
            ex Z being set st y in Z & Z in f"F0
            proof
              set Z = f"Q;
              take Z;
              thus thesis by A10,A9,FUNCT_1:def 7,FUNCT_2:def 9;
            end;
            hence thesis by TARSKI:def 4;
          end;
        end;
        assume
A11:    x in P;
        then
A12:    f.x in f.:P by A3,FUNCT_1:def 6;
        reconsider x as Point of T by A11;
A13:    f.x in union F0 by A6,A12;
A14:    f"{f.x} c= f"(union F0)
        proof
          let y be object;
          assume
A15:      y in f"{f.x};
          then f.y in {f.x} by FUNCT_1:def 7;
          then
A16:      f.y in union F0 by A13,TARSKI:def 1;
          y in dom f by A15,FUNCT_1:def 7;
          hence thesis by A16,FUNCT_1:def 7;
        end;
        f.x in {f.x} by TARSKI:def 1;
        then x in f"{f.x} by A3,A11,FUNCT_1:def 7;
        then x in f"(union F0) by A14;
        hence thesis by A7;
      end;
    end;
    then
A17: B0 is Cover of P by SETFAM_1:def 11;
    B0 is open by A2,A5,Th4;
    then ex B being Subset-Family of T st B c= B0 & B is Cover of P & B is
    finite by A1,A17,COMPTS_1:def 4;
    then consider G being Subset-Family of S such that
A18: G c= F0 & G is Cover of f.:P & G is finite by Th7;
    take G;
    thus thesis by A18;
  end;
  hence thesis by COMPTS_1:def 4;
end;
