let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M
for x being set holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for W1, W2 being Subspace of M
for x being set holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
let W1, W2 be Subspace of M; :: thesis: for x being set holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
let x be set ; :: 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 Def2;
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