
theorem Th14:
  for X being set, A being Subset-Family of X st X in A
  for x being set holds x in FinMeetCl A iff
  ex Y being finite non empty Subset-Family of X st Y c= A & x = Intersect Y
proof
  let X be set, A be Subset-Family of X;
  assume
A1: X in A;
  then
A2: {X} c= A by ZFMISC_1:31;
  reconsider Z = {X} as finite non empty Subset-Family of X by A1,ZFMISC_1:31;
  reconsider Z as finite non empty Subset-Family of X;
A3: Intersect Z = meet Z by SETFAM_1:def 9
    .= X by SETFAM_1:10;
  let x be set;
  hereby
    assume x in FinMeetCl A;
    then consider Y being Subset-Family of X such that
A4: Y c= A and
A5: Y is finite and
A6: x = Intersect Y by CANTOR_1:def 3;
    per cases;
    suppose Y = {};
      then x = X by A6,SETFAM_1:def 9;
      hence ex Y being finite non empty Subset-Family of X st
      Y c= A & x = Intersect Y by A2,A3;
    end;
    suppose Y <> {};
      hence ex Y being finite non empty Subset-Family of X st
      Y c= A & x = Intersect Y by A4,A5,A6;
    end;
  end;
  thus thesis by CANTOR_1:def 3;
end;
