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

proof

assume W1 is Submodule of W2 ; :: thesis: W1 /\ W2 = W1
then A1: the carrier of W1 c= the carrier of W2 by Def9;
the carrier of (W1 /\ W2) = the carrier of W1 /\ the carrier of W2 by Def15;
hence W1 /\ W2 = W1 by A1, Th45, XBOOLE_1:28; :: thesis: verum

end;

thus ( W1 /\ W2 = W1 implies W1 is Submodule of W2 ) by Th105; :: thesis: verum