now :: thesis: for x being object holds
( x in {1,2,3} \ {2} iff x in {1,3} )

let x be object ; :: thesis: ( x in {1,2,3} \ {2} iff x in {1,3} )
( x in {1,2,3} \ {2} iff ( x in {1,2,3} & not x in {2} ) ) by XBOOLE_0:def 5;
then ( x in {1,2,3} \ {2} iff ( ( x = 1 or x = 2 or x = 3 ) & not x = 2 ) ) by ENUMSET1:def 1, TARSKI:def 1;
hence ( x in {1,2,3} \ {2} iff x in {1,3} ) by TARSKI:def 2; :: thesis: verum

end;

hence {1,2,3} \ {2} = {1,3} by TARSKI:2; :: thesis: verum