let R be Ring; :: thesis:
let V be RightMod of R; :: thesis:
let W1, W2, W3 be Submodule of V; :: thesis:
assume A1: ( W1 is Submodule of W3 & W2 is Submodule of W3 ) ; :: thesis:

let v be Vector of V; :: thesis:
assume v in W1 + W2 ; :: thesis:
then consider v1, v2 being Vector of V such that
A2: ( v1 in W1 & v2 in W2 ) and
A3: v = v1 + v2 by Th1;
( v1 in W3 & v2 in W3 ) by A1, A2, RMOD_2:8;
hence v in W3 by A3, RMOD_2:20; :: thesis:

end;