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
  [:X,Y:] c= bool bool (X \/ Y)
proof
  let z;
   reconsider zz=z as set by TARSKI:1;
  assume z in [:X,Y:];
  then consider x,y such that
A1: x in X and
A2: y in Y and
A3: z = [x,y] by Def2;
  zz c= bool (X \/ Y)
  proof
    let u;
    assume u in zz;
    then
A4: u = {x,y} or u = {x} by A3,TARSKI:def 2;
    X c= X \/ Y & {x} c= X by A1,Lm1,XBOOLE_1:7;
    then
A5: {x} c= X \/ Y;
    x in X \/ Y & y in X \/ Y by A1,A2,XBOOLE_0:def 3;
    then {x,y} c= X \/ Y by Th31;
    hence thesis by A5,A4,Def1;
  end;
  hence thesis by Def1;
end;
