reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem
  A c= B implies bool A c= bool B
proof
  assume
A1: A c= B;
  let x;
   reconsider xx=x as set by TARSKI:1;
  assume x in bool A;
  then xx c= A by Def1;
  then xx c= B by A1;
  hence thesis by Def1;
end;
