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

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

W1 is Submodule of W2 + W3

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

W1 is Submodule of W2 + W3

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

A1: W2 is Submodule of W2 + W3 by ZMODUL01:97;

assume W1 is Submodule of W2 ; :: thesis: W1 is Submodule of W2 + W3

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

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

W1 is Submodule of W2 + W3

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

W1 is Submodule of W2 + W3

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

A1: W2 is Submodule of W2 + W3 by ZMODUL01:97;

assume W1 is Submodule of W2 ; :: thesis: W1 is Submodule of W2 + W3

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