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;
reserve e for object, X,X1,X2,Y1,Y2 for set;

theorem
  N c= [:X1,Y1:] & M c= [:X2,Y2:] implies N \/ M c= [:X1 \/ X2,Y1 \/ Y2 :]
proof
  assume N c= [:X1,Y1:] & M c= [:X2,Y2:];
  then
A1: N \/ M c= [:X1,Y1:] \/ [:X2,Y2:] by XBOOLE_1:13;
  X1 c= X1 \/ X2 & Y1 c= Y1 \/ Y2 by XBOOLE_1:7;
  then
A2: [:X1,Y1:] c= [:X1 \/ X2,Y1 \/ Y2:] by Th95;
  Y2 c= Y1 \/ Y2 & X2 c= X1 \/ X2 by XBOOLE_1:7;
  then [:X2,Y2:] c= [:X1 \/ X2,Y1 \/ Y2:] by Th95;
  then [:X1,Y1:] \/ [:X2,Y2:] c= [:X1 \/ X2,Y1 \/ Y2:] by A2,XBOOLE_1:8;
  hence thesis by A1;
end;
