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 )

( 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

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