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 Th84:
  z in [:X1, Y1:] /\ [:X2, Y2:] implies ex x,y st z = [x,y] & x
  in X1 /\ X2 & y in Y1 /\ Y2
proof
  assume
A1: z in [:X1, Y1:] /\ [:X2, Y2:];
  then z in [:X1, Y1:] by XBOOLE_0:def 4;
  then consider x1,y1 such that
A2: x1 in X1 and
A3: y1 in Y1 and
A4: z=[x1,y1] by Def2;
  z in [:X2, Y2:] by A1,XBOOLE_0:def 4;
  then consider x2,y2 such that
A5: x2 in X2 and
A6: y2 in Y2 and
A7: z=[x2,y2] by Def2;
  y1=y2 by A4,A7,XTUPLE_0:1;
  then
A8: y1 in Y1 /\ Y2 by A3,A6,XBOOLE_0:def 4;
  x1=x2 by A4,A7,XTUPLE_0:1;
  then x1 in X1 /\ X2 by A2,A5,XBOOLE_0:def 4;
  hence thesis by A4,A8;
end;
