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 & W1 is Submodule of W3 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 & 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

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 LeftMod 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

hence
W1 is Submodule of W2 /\ W3
by ZMODUL01:44; :: thesis: verumv 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, ZMODUL01:23;

hence v in W2 /\ W3 by ZMODUL01:94; :: thesis: verum

end;assume v in W1 ; :: thesis: v in W2 /\ W3

then ( v in W2 & v in W3 ) by A1, ZMODUL01:23;

hence v in W2 /\ W3 by ZMODUL01:94; :: thesis: verum