reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th23:
  for F being Subset-Family of X, H being Subset-Family of [:X,Y:]
st F is finite & F c= Pr1(X,Y).:H ex G being Subset-Family of [:X,Y:] st G c= H
  & G is finite & F = Pr1(X,Y).:G
proof
  let F be Subset-Family of X, H be Subset-Family of [:X,Y:] such that
A1: F is finite and
A2: F c= Pr1(X,Y).:H;
  defpred P[object,object] means
   $2 in dom Pr1(X,Y) & $2 in H & $1 = Pr1(X,Y).($2);
A3: for e being object st e in F holds
     ex u being object st P[e,u] by A2,FUNCT_1:def 6;
  consider f being Function such that
A4: dom f = F and
A5: for e being object st e in F holds P[e,f.e] from CLASSES1:sch 1(A3);
A6: f.:F c= H
  proof
    let e be object;
    assume e in f.:F;
    then ex u being object st u in dom f & u in F & e = f.u by FUNCT_1:def 6;
    hence thesis by A5;
  end;
  then reconsider G = f.:F as Subset-Family of [:X,Y:] by XBOOLE_1:1;
  take G;
  thus G c= H by A6;
  thus G is finite by A1;
  now
    let e be object;
    thus e in F iff
     ex u being object st u in dom Pr1(X,Y) & u in G & e = Pr1(X,Y).u
    proof
      thus e in F implies
       ex u being object st u in dom Pr1(X,Y) & u in G & e = Pr1(X,Y).u
      proof
        assume
A7:     e in F;
        take f.e;
        thus f.e in dom Pr1(X,Y) by A5,A7;
        thus f.e in G by A4,A7,FUNCT_1:def 6;
        thus thesis by A5,A7;
      end;
      given u being object such that
      u in dom Pr1(X,Y) and
A8:   u in G and
A9:   e = Pr1(X,Y).u;
      ex v being object st v in dom f & v in F & u = f.v by A8,FUNCT_1:def 6;
      hence thesis by A5,A9;
    end;
  end;
  hence thesis by FUNCT_1:def 6;
end;
