let X, Y be set ; :: thesis: (proj1_4 X) \ (proj1_4 Y) c= proj1_4 (X \ Y)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (proj1_4 X) \ (proj1_4 Y) or x in proj1_4 (X \ Y) )
assume A1: x in (proj1_4 X) \ (proj1_4 Y) ; :: thesis: x in proj1_4 (X \ Y)
then x in proj1_4 X by XBOOLE_0:def 5;
then consider x1, x2, x3 being object 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 x in proj1_4 (X \ Y) by Th19; :: thesis: verum