let R be Ring; :: thesis: for V being LeftMod of R

for W1, W2, W3 being Submodule of V st W1 is Submodule of W3 holds

W1 /\ W2 is Submodule of W3

let V be LeftMod of R; :: thesis: for W1, W2, W3 being Submodule of V st W1 is Submodule of W3 holds

W1 /\ W2 is Submodule of W3

let W1, W2, W3 be Submodule of V; :: thesis: ( W1 is Submodule of W3 implies W1 /\ W2 is Submodule of W3 )

A1: W1 /\ W2 is Submodule of W1 by ZMODUL01:105;

assume W1 is Submodule of W3 ; :: thesis: W1 /\ W2 is Submodule of W3

hence W1 /\ W2 is Submodule of W3 by A1, ZMODUL01:42; :: thesis: verum

for W1, W2, W3 being Submodule of V st W1 is Submodule of W3 holds

W1 /\ W2 is Submodule of W3

let V be LeftMod of R; :: thesis: for W1, W2, W3 being Submodule of V st W1 is Submodule of W3 holds

W1 /\ W2 is Submodule of W3

let W1, W2, W3 be Submodule of V; :: thesis: ( W1 is Submodule of W3 implies W1 /\ W2 is Submodule of W3 )

A1: W1 /\ W2 is Submodule of W1 by ZMODUL01:105;

assume W1 is Submodule of W3 ; :: thesis: W1 /\ W2 is Submodule of W3

hence W1 /\ W2 is Submodule of W3 by A1, ZMODUL01:42; :: thesis: verum