reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem Th12:
  for A being non empty set st for a being Element of A holds varcl a = a
  holds varcl meet A = meet A
proof
  let B be non empty set;
  set A = meet B;
  assume
A1: for a being Element of B holds varcl a = a;
  now
    thus A c= A;
    let x,y;
    assume
A2: [x,y] in A;
    now
      let Y;
      assume
A3:   Y in B;
      then
A4:   [x,y] in Y by A2,SETFAM_1:def 1;
      Y = varcl Y by A1,A3;
      hence x c= Y by A4,Def1;
    end;
    hence x c= A by SETFAM_1:5;
  end;
  hence varcl A c= A by Def1;
  thus thesis by Def1;
end;
