let V be strict Z_Module; :: thesis: for W being strict Submodule of V st ( for v being VECTOR of V holds
( v in W iff v in V ) ) holds
W = V
let W be strict Submodule of V; :: thesis: ( ( for v being VECTOR of V holds
( v in W iff v in V ) ) implies W = V )
assume A1: for v being VECTOR of V holds
( v in W iff v in V ) ; :: thesis: W = V
V is Submodule of V by Th32;
hence W = V by A1, Th46; :: thesis: verum