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 Th99:
  [:X1 /\ X2, Y1 /\ Y2:] = [:X1,Y1:] /\ [:X2, Y2:]
proof
  Y1 /\ Y2 c= Y2 & X1 /\ X2 c= X2 by XBOOLE_1:17;
  then
A1: [:X1 /\ X2, Y1 /\ Y2:] c= [:X2, Y2:] by Th95;
A2: [:X1, Y1:] /\ [:X2, Y2:] c= [:X1 /\ X2, Y1 /\ Y2:]
  proof
    let z;
    assume z in [:X1, Y1:] /\ [:X2, Y2:];
    then ex x,y st z=[x,y] & x in X1 /\ X2 & y in Y1 /\ Y2 by Th84;
    hence thesis by Def2;
  end;
  Y1 /\ Y2 c= Y1 & X1 /\ X2 c= X1 by XBOOLE_1:17;
  then [:X1 /\ X2, Y1 /\ Y2:] c= [:X1, Y1:] by Th95;
  hence thesis by A2,A1,XBOOLE_1:19;
end;
