reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;

theorem
  F is finite implies F|M is finite
proof
  defpred X[object,object] means
  for X being Subset of T st X = $1 holds $2 = X /\ M;
A1: for x being object st x in F ex y being object st X[x,y]
  proof
    let x be object;
    assume x in F;
    then reconsider X=x as Subset of T;
    reconsider y=X /\ M as set;
    take y;
    thus thesis;
  end;
  consider f being Function such that
A2: dom f = F and
A3: for x being object st x in F holds X[x,f.x] from CLASSES1:sch 1(A1);
  for x being object holds x in rng f iff x in F|M
  proof let x be object;
    hereby
      assume x in rng f;
      then consider y being object such that
A4:   y in dom f and
A5:   x=f.y by FUNCT_1:def 3;
      reconsider Y=y as Subset of T by A2,A4;
      Y /\ M c= M by XBOOLE_1:17;
      then Y /\ M c= [#](T|M) by PRE_TOPC:def 5;
      then reconsider X=f.y as Subset of T|M by A2,A3,A4;
      f.y = Y /\ M by A2,A3,A4;
      then X in F|M by A2,A4,Def3;
      hence x in F|M by A5;
    end;
    assume
A6: x in F|M;
    then reconsider X=x as Subset of T|M;
    consider R being Subset of T such that
A7: R in F and
A8: R /\ M = X by A6,Def3;
    f.R = R /\ M by A3,A7;
    hence thesis by A2,A7,A8,FUNCT_1:def 3;
  end;
  then rng f = F|M by TARSKI:2;
  then
A9: f.:(F) = F|M by A2,RELAT_1:113;
  assume F is finite;
  hence thesis by A9,FINSET_1:5;
end;
