let V be Z_Module; :: thesis: for W1 being Submodule of V
for W2 being strict Submodule of V holds
( W1 is Submodule of W2 iff W1 + W2 = W2 )
let W1 be Submodule of V; :: thesis: for W2 being strict Submodule of V holds
( W1 is Submodule of W2 iff W1 + W2 = W2 )
let W2 be strict Submodule of V; :: thesis: ( W1 is Submodule of W2 iff W1 + W2 = W2 )
thus ( W1 is Submodule of W2 implies W1 + W2 = W2 ) :: thesis: ( W1 + W2 = W2 implies W1 is Submodule of W2 )

proof

assume W1 is Submodule of W2 ; :: thesis: W1 + W2 = W2
then the carrier of W1 c= the carrier of W2 by Def9;
hence W1 + W2 = W2 by Lm7; :: thesis: verum

end;

thus ( W1 + W2 = W2 implies W1 is Submodule of W2 ) by Th97; :: thesis: verum