let V be Z_Module; :: thesis: for v being VECTOR of V
for W being Submodule of V holds
( 0. V in v + W iff v + W = the carrier of W )
let v be VECTOR of V; :: thesis: for W being Submodule of V holds
( 0. V in v + W iff v + W = the carrier of W )
let W be Submodule of V; :: thesis: ( 0. V in v + W iff v + W = the carrier of W )
( 0. V in v + W iff v in W ) by Th57;
hence ( 0. V in v + W iff v + W = the carrier of W ) by Lm5; :: thesis: verum