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 Th45:
  proj2_4 X \ proj2_4 Y c= proj2_4(X \ Y)
proof
 let x be object;
 assume
A1: x in proj2_4 X \ proj2_4 Y;
  then x in proj2_4 X by XBOOLE_0:def 5;
  then consider x1,x2,x3 such that
A2: [x1,x,x2,x3] in X by Th20;
  not x in proj2_4 Y by A1,XBOOLE_0:def 5;
  then not [x1,x,x2,x3] in Y by Th21;
  then [x1,x,x2,x3] in X \ Y by A2,XBOOLE_0:def 5;
 hence thesis by Th21;
end;
