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 Th41:
  proj1_4 X \ proj1_4 Y c= proj1_4(X \ Y)
proof
 let x be object;
 assume
A1: x in proj1_4 X \ proj1_4 Y;
  then x in proj1_4 X by XBOOLE_0:def 5;
  then consider x1,x2,x3 such that
A2: [x,x1,x2,x3] in X by Th18;
  not x in proj1_4 Y by A1,XBOOLE_0:def 5;
  then not [x,x1,x2,x3] in Y by Th19;
  then [x,x1,x2,x3] in X \ Y by A2,XBOOLE_0:def 5;
 hence thesis by Th19;
end;
