
theorem Th20:
  for X,Y being set, A being Subset-Family of X, B being Subset-Family of Y
  st A = B & X in A
  holds FinMeetCl B = {Y} \/ FinMeetCl A
proof
  let X,Y be set, A be Subset-Family of X, B be Subset-Family of Y such that
A1: A = B and
A2: X in A;
  thus FinMeetCl B c= {Y} \/ FinMeetCl A
  proof
    let x be object;
    assume x in FinMeetCl B;
    then consider y being Subset-Family of Y such that
A3: y c= B and
A4: y is finite and
A5: x = Intersect y by CANTOR_1:def 3;
    reconsider z = y as Subset-Family of X by A1,A3,XBOOLE_1:1;
    reconsider z as Subset-Family of X;
    per cases;
    suppose y = {};
      then x = Y by A5,SETFAM_1:def 9;
      then x in {Y} by TARSKI:def 1;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
A6:   y <> {};
      then
A7:   Intersect y = meet y by SETFAM_1:def 9;
      Intersect z = meet y by A6,SETFAM_1:def 9;
      then x in FinMeetCl A by A1,A3,A4,A5,A7,CANTOR_1:def 3;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  let x be object;
  assume
A8: x in {Y} \/ FinMeetCl A;
  per cases by A8,XBOOLE_0:def 3;
  suppose x in {Y};
    then
A9: x = Y by TARSKI:def 1;
A10: Intersect {}bool Y = Y by SETFAM_1:def 9;
    {}bool Y c= B;
    hence thesis by A9,A10,CANTOR_1:def 3;
  end;
  suppose x in FinMeetCl A;
    then consider y being non empty finite Subset-Family of X such that
A11: y c= A and
A12: x = Intersect y by A2,Th14;
    reconsider z = y as Subset-Family of Y by A1,A11,XBOOLE_1:1;
    reconsider z as Subset-Family of Y;
    x = meet z by A12,SETFAM_1:def 9
      .= Intersect z by SETFAM_1:def 9;
    hence thesis by A1,A11,CANTOR_1:def 3;
  end;
end;
