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
  [:X1 \/ X2, Y1 \/ Y2:] = [:X1,Y1:] \/ [:X1,Y2:] \/ [:X2,Y1:] \/ [:X2, Y2:]
proof
  thus [:X1 \/ X2, Y1 \/ Y2:] = [:X1, Y1 \/ Y2:] \/ [:X2, Y1 \/ Y2:] by Th96
    .= [:X1,Y1:] \/ [:X1,Y2:] \/ [:X2,Y1 \/ Y2:] by Th96
    .= [:X1,Y1:] \/ [:X1,Y2:] \/ ([:X2,Y1:] \/ [:X2,Y2:]) by Th96
    .= [:X1,Y1:] \/ [:X1,Y2:] \/ [:X2,Y1:] \/ [:X2,Y2:] by XBOOLE_1:4;
end;
