let x be object ; :: thesis: for V being RealLinearSpace holds
( x in (0). V iff x = 0. V )
let V be RealLinearSpace; :: thesis: ( x in (0). V iff x = 0. V )
thus ( x in (0). V implies x = 0. V ) :: thesis: ( x = 0. V implies x in (0). V )

proof

assume x in (0). V ; :: thesis: x = 0. V
then x in the carrier of ((0). V) by STRUCT_0:def 5;
then x in {(0. V)} by RLSUB_1:def 3;
hence x = 0. V by TARSKI:def 1; :: thesis: verum

end;

thus ( x = 0. V implies x in (0). V ) by RLSUB_1:17; :: thesis: verum