let V be RealUnitarySpace; :: thesis: for x being object holds
( x in (0). V iff x = 0. V )
let x be object ; :: 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 RUSUB_1:def 2;
hence x = 0. V by TARSKI:def 1; :: thesis: verum

end;

thus ( x = 0. V implies x in (0). V ) by RUSUB_1:11; :: thesis: verum