let x be set ; :: thesis: for V being Z_Module
for W1, W2 being Submodule of V holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
let V be Z_Module; :: thesis: for W1, W2 being Submodule of V holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
let W1, W2 be Submodule of V; :: thesis: ( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
( x in W1 /\ W2 iff x in the carrier of (W1 /\ W2) ) by STRUCT_0:def 5;
then ( x in W1 /\ W2 iff x in the carrier of W1 /\ the carrier of W2 ) by Def15;
then ( x in W1 /\ W2 iff ( x in the carrier of W1 & x in the carrier of W2 ) ) by XBOOLE_0:def 4;
hence ( x in W1 /\ W2 iff ( x in W1 & x in W2 ) ) by STRUCT_0:def 5; :: thesis: verum