reserve x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2,z2,z4 for object;
reserve X,X1,X2,X3,X4,Y for set;

theorem
  proj1_3 X \+\ proj1_3 Y c= proj1_3(X \+\ Y)
proof
  proj1_3 X \ proj1_3 Y c= proj1_3(X \ Y) &
  proj1_3 Y \ proj1_3 X c= proj1_3(Y \ X) by Th33;
  then proj1_3 X \+\ proj1_3 Y c= proj1_3(X\Y) \/ proj1_3(Y\X) by XBOOLE_1:13;
 hence thesis by Th31;
end;
