let V be RealLinearSpace; :: thesis: for W1, W2, W3 being Subspace of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))

let W1, W2, W3 be Subspace of V; :: thesis: the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) or x in the carrier of (W2 /\ (W1 + W3)) )

assume x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) ; :: thesis: x in the carrier of (W2 /\ (W1 + W3))

then x in { (u + v) where v, u is VECTOR of V : ( u in W1 /\ W2 & v in W2 /\ W3 ) } by Def1;

then consider v, u being VECTOR of V such that

A1: x = u + v and

A2: ( u in W1 /\ W2 & v in W2 /\ W3 ) ;

( u in W2 & v in W2 ) by A2, Th3;

then A3: x in W2 by A1, RLSUB_1:20;

( u in W1 & v in W3 ) by A2, Th3;

then x in W1 + W3 by A1, Th1;

then x in W2 /\ (W1 + W3) by A3, Th3;

hence x in the carrier of (W2 /\ (W1 + W3)) by STRUCT_0:def 5; :: thesis: verum

let W1, W2, W3 be Subspace of V; :: thesis: the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) or x in the carrier of (W2 /\ (W1 + W3)) )

assume x in the carrier of ((W1 /\ W2) + (W2 /\ W3)) ; :: thesis: x in the carrier of (W2 /\ (W1 + W3))

then x in { (u + v) where v, u is VECTOR of V : ( u in W1 /\ W2 & v in W2 /\ W3 ) } by Def1;

then consider v, u being VECTOR of V such that

A1: x = u + v and

A2: ( u in W1 /\ W2 & v in W2 /\ W3 ) ;

( u in W2 & v in W2 ) by A2, Th3;

then A3: x in W2 by A1, RLSUB_1:20;

( u in W1 & v in W3 ) by A2, Th3;

then x in W1 + W3 by A1, Th1;

then x in W2 /\ (W1 + W3) by A3, Th3;

hence x in the carrier of (W2 /\ (W1 + W3)) by STRUCT_0:def 5; :: thesis: verum