let R be Ring; :: thesis: for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 & W1 is Submodule of W3 holds
W1 is Submodule of W2 /\ W3
let V be RightMod of R; :: thesis: for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 & W1 is Submodule of W3 holds
W1 is Submodule of W2 /\ W3
let W1, W2, W3 be Submodule of V; :: thesis: ( W1 is Submodule of W2 & W1 is Submodule of W3 implies W1 is Submodule of W2 /\ W3 )
assume A1: ( W1 is Submodule of W2 & W1 is Submodule of W3 ) ; :: thesis: W1 is Submodule of W2 /\ W3
now :: thesis: for v being Vector of V st v in W1 holds
v in W2 /\ W3
let v be Vector of V; :: thesis: ( v in W1 implies v in W2 /\ W3 )
assume v in W1 ; :: thesis: v in W2 /\ W3
then ( v in W2 & v in W3 ) by A1, RMOD_2:8;
hence v in W2 /\ W3 by Th3; :: thesis: verum
end;
hence W1 is Submodule of W2 /\ W3 by RMOD_2:28; :: thesis: verum