
theorem Th7:
  for T,S being non empty TopSpace holds for f being Function of T
,S holds for P being Subset of T holds for F being Subset-Family of S holds (ex
  B being Subset-Family of T st (B c= f"F & B is Cover of P & B is finite))
  implies ex G being Subset-Family of S st G c= F & G is Cover of f.:P & G is
  finite
proof
  let T,S be non empty TopSpace;
  let f be Function of T,S;
  let P be Subset of T;
  let F be Subset-Family of S;
  given B being Subset-Family of T such that
A1: B c= f"F and
A2: B is Cover of P and
A3: B is finite;
A4: P c= union B by A2,SETFAM_1:def 11;
  now
    per cases;
    case
A5:   P <> {};
      thus ex G being Subset-Family of S st G c= F & G is Cover of f.:P &
      G is finite
      proof
        consider s being FinSequence such that
A6:     rng s = B by A3,FINSEQ_1:52;
        B <> {} by A4,A5,ZFMISC_1:2;
        then consider D being non empty set such that
A7:     D = B;
        defpred P0[Element of D,Subset of [#](S)] means for x being Element of
D st $1 = x holds for y being Subset of [#](S) st $2 = y holds (y in F & x = f"
        y);
A8:     for x being Element of D ex y being Subset of [#](S) st P0[x,y]
        proof
          let x be Element of D;
A9:       x in B by A7;
          then reconsider x as Subset of T;
          consider y being Subset of S such that
A10:      y in F & x = f"y by A1,A9,FUNCT_2:def 9;
          reconsider y as Subset of [#](S);
          take y;
          thus thesis by A10;
        end;
        consider F0 being Function of D,bool [#](S) such that
A11:    for x being Element of D holds P0[x,F0.x qua Subset of [#](S)
        ] from FUNCT_2:sch 3(A8);
A12:    for x being Element of D holds F0.x in F & x = f"(F0.x) by A11;
        reconsider F0 as Function of B,bool [#](S) by A7;
A13:    dom F0 = B by FUNCT_2:def 1;
        then reconsider q = F0*s as FinSequence by A6,FINSEQ_1:16;
        set G = rng q;
A14:    G c= F
        proof
          let x be object;
          assume x in G;
          then consider y being object such that
A15:      y in dom q & x = q.y by FUNCT_1:def 3;
          s.y in B & x = F0.(s.y) by A13,A15,FUNCT_1:11,12;
          hence thesis by A7,A12;
        end;
        then reconsider G as Subset-Family of S by XBOOLE_1:1;
        reconsider G as Subset-Family of S;
        take G;
        for x being object holds x in f.:P implies x in union G
        proof
          let x be object;
          assume
A16:      x in f.:P;
          ex A being set st x in A & A in G
          proof
            consider y being object such that
A17:        y in dom f and
A18:        y in P and
A19:        x = f.y by A16,FUNCT_1:def 6;
            consider C being set such that
A20:        y in C and
A21:        C in B by A4,A18,TARSKI:def 4;
            C = f"(F0.C) by A7,A12,A21;
            then
A22:        x in f.:(f"(F0.C)) by A17,A19,A20,FUNCT_1:def 6;
            set A = F0.C;
            take A;
            f.:(f"(F0.C)) c= F0.C & G = rng F0 by A6,A13,FUNCT_1:75,RELAT_1:28;
            hence thesis by A21,A22,FUNCT_2:4;
          end;
          hence thesis by TARSKI:def 4;
        end;
        then f.:P c= union G;
        hence thesis by A14,SETFAM_1:def 11;
      end;
      hence thesis;
    end;
    case
A23:  P = {};
      ex G being Subset-Family of S st G c= F & G is Cover of f.:P & G is
      finite
      proof
        set G = {};
        reconsider G as Subset-Family of S by XBOOLE_1:2;
        reconsider G as Subset-Family of S;
        take G;
        f.:P = {}
        proof
          assume f.:P <> {};
          then consider x being object such that
A24:      x in f.:P by XBOOLE_0:def 1;
          ex z being object st z in dom f & z in P & x = f.z
               by A24,FUNCT_1:def 6;
          hence contradiction by A23;
        end;
        hence thesis by SETFAM_1:def 11,ZFMISC_1:2;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
