
theorem
  for T,S being non empty TopSpace holds for f being Function of T,S
  holds for P being Subset-Family of T holds P is finite implies f.:P is finite
proof
  let T,S be non empty TopSpace;
  let f be Function of T,S;
  let P be Subset-Family of T;
  defpred EF[Subset of [#](T),Subset of [#](S)] means for s,t being set holds
  ($1 = s & $2 = t implies t = f.:s);
  assume P is finite;
  then consider s being FinSequence such that
A1: rng s = P by FINSEQ_1:52;
A2: for x being Subset of [#](T) ex y being Subset of [#](S) st EF[x,y]
  proof
    let x be Subset of [#](T);
    reconsider x as set;
    set y = f.:x;
    reconsider y as Subset of [#](S);
    take y;
    thus thesis;
  end;
  consider F being Function of bool [#](T),bool [#](S) such that
A3: for x being Subset of [#](T) holds EF[x,F.x qua Subset of [#](S)]
  from FUNCT_2:sch 3 (A2);
  dom F = bool [#](T) by FUNCT_2:def 1;
  then reconsider q = F*s as FinSequence by A1,FINSEQ_1:16;
  for x being object holds x in F.:P iff x in f.:P
  proof
    let x be object;
    thus x in F.:P implies x in f.:P
    proof
      assume x in F.:P;
      then consider y being object such that
A4:   y in dom F and
A5:   y in P & x = F.y by FUNCT_1:def 6;
      reconsider y as Subset of T by A4;
      F.y = f.:y by A3;
      hence thesis by A5,FUNCT_2:def 10;
    end;
    thus x in f.:P implies x in F.:P
    proof
      assume
A6:   x in f.:P;
      then reconsider x as Subset of S;
      consider y being Subset of T such that
A7:   y in P and
A8:   x = f.:y by A6,FUNCT_2:def 10;
A9:   dom F = bool [#](T) by FUNCT_2:def 1;
      x = F.y by A3,A8;
      hence thesis by A7,A9,FUNCT_1:def 6;
    end;
  end;
  then
A10: F.:P = f.:P by TARSKI:2;
  ex q being FinSequence st rng q = f.:P
  proof
    take q;
    thus thesis by A1,A10,RELAT_1:127;
  end;
  hence thesis;
end;
