let R be Ring; :: thesis: for V being RightMod of R
for W1, W2 being Submodule of V holds
( W1 is Submodule of W1 + W2 & W2 is Submodule of W1 + W2 )
let V be RightMod of R; :: thesis: for W1, W2 being Submodule of V holds
( W1 is Submodule of W1 + W2 & W2 is Submodule of W1 + W2 )
let W1, W2 be Submodule of V; :: thesis: ( W1 is Submodule of W1 + W2 & W2 is Submodule of W1 + W2 )
the carrier of W1 c= the carrier of (W1 + W2) by Lm2;
hence W1 is Submodule of W1 + W2 by RMOD_2:27; :: thesis: W2 is Submodule of W1 + W2
the carrier of W2 c= the carrier of (W2 + W1) by Lm2;
then the carrier of W2 c= the carrier of (W1 + W2) by Lm1;
hence W2 is Submodule of W1 + W2 by RMOD_2:27; :: thesis: verum