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 Th37:
  proj2_3 X \ proj2_3 Y c= proj2_3(X \ Y)
proof
 let x be object;
 assume
A1: x in proj2_3 X \ proj2_3 Y;
  then x in proj2_3 X by XBOOLE_0:def 5;
  then consider y,z such that
A2: [y,x,z] in X by Th14;
  not x in proj2_3 Y by A1,XBOOLE_0:def 5;
  then not [y,x,z] in Y by Th15;
  then [y,x,z] in X \ Y by A2,XBOOLE_0:def 5;
 hence thesis by Th15;
end;
