let R be Ring; :: thesis: for V being RightMod of R
for W2 being Submodule of V holds
( ( for W1 being strict Submodule of V st W1 is Submodule of W2 holds
W1 /\ W2 = W1 ) & ( for W1 being Submodule of V st W1 /\ W2 = W1 holds
W1 is Submodule of W2 ) )
let V be RightMod of R; :: thesis: for W2 being Submodule of V holds
( ( for W1 being strict Submodule of V st W1 is Submodule of W2 holds
W1 /\ W2 = W1 ) & ( for W1 being Submodule of V st W1 /\ W2 = W1 holds
W1 is Submodule of W2 ) )
let W2 be Submodule of V; :: thesis: ( ( for W1 being strict Submodule of V st W1 is Submodule of W2 holds
W1 /\ W2 = W1 ) & ( for W1 being Submodule of V st W1 /\ W2 = W1 holds
W1 is Submodule of W2 ) )
thus for W1 being strict Submodule of V st W1 is Submodule of W2 holds
W1 /\ W2 = W1 :: thesis: for W1 being Submodule of V st W1 /\ W2 = W1 holds
W1 is Submodule of W2
proof
let W1 be strict Submodule of V; :: thesis: ( W1 is Submodule of W2 implies W1 /\ W2 = W1 )
assume W1 is Submodule of W2 ; :: thesis: W1 /\ W2 = W1
then A1: the carrier of W1 c= the carrier of W2 by RMOD_2:def 2;
the carrier of (W1 /\ W2) = the carrier of W1 /\ the carrier of W2 by Def2;
hence W1 /\ W2 = W1 by A1, RMOD_2:29, XBOOLE_1:28; :: thesis: verum
end;
thus for W1 being Submodule of V st W1 /\ W2 = W1 holds
W1 is Submodule of W2 by Th16; :: thesis: verum