let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V

for x being object holds

( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )

let W1, W2 be Subspace of V; :: thesis: for x being object holds

( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )

let x be object ; :: 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

for x being object holds

( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )

let W1, W2 be Subspace of V; :: thesis: for x being object holds

( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )

let x be object ; :: 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