let F be non degenerated almost_left_invertible Ring; :: thesis: for x being Scalar of F
for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of F
for v being Vector of V holds
( v * x = 0. V iff ( x = 0. F or v = 0. V ) )
let x be Scalar of F; :: thesis: for V being non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of F
for v being Vector of V holds
( v * x = 0. V iff ( x = 0. F or v = 0. V ) )
let V be non empty right_complementable add-associative right_zeroed RightMod-like RightModStr of F; :: thesis: for v being Vector of V holds
( v * x = 0. V iff ( x = 0. F or v = 0. V ) )
let v be Vector of V; :: thesis: ( v * x = 0. V iff ( x = 0. F or v = 0. V ) )
( not v * x = 0. V or x = 0. F or v = 0. V )

proof

assume v * x = 0. V ; :: thesis: ( x = 0. F or v = 0. V )
then A1: v * (x * (x ")) = (0. V) * (x ") by Def23
.= 0. V by Th90 ;
assume x <> 0. F ; :: thesis: v = 0. V
then 0. V = v * (1_ F) by A1, Th43;
hence v = 0. V by Def23; :: thesis: verum

end;

hence ( v * x = 0. V iff ( x = 0. F or v = 0. V ) ) by Th90; :: thesis: verum