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

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

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

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

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

then consider v, u being VECTOR of V such that

A1: ( x = u + v & u in W2 ) and

A2: v in W1 /\ W3 ;

v in W3 by A2, Th3;

then x in { (u1 + u2) where u2, u1 is VECTOR of V : ( u1 in W2 & u2 in W3 ) } by A1;

then A3: x in the carrier of (W2 + W3) by Def1;

v in W1 by A2, Th3;

then x in { (v1 + v2) where v2, v1 is VECTOR of V : ( v1 in W1 & v2 in W2 ) } by A1;

then x in the carrier of (W1 + W2) by Def1;

then x in the carrier of (W1 + W2) /\ the carrier of (W2 + W3) by A3, XBOOLE_0:def 4;

hence x in the carrier of ((W1 + W2) /\ (W2 + W3)) by Def2; :: thesis: verum

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

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

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

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

then consider v, u being VECTOR of V such that

A1: ( x = u + v & u in W2 ) and

A2: v in W1 /\ W3 ;

v in W3 by A2, Th3;

then x in { (u1 + u2) where u2, u1 is VECTOR of V : ( u1 in W2 & u2 in W3 ) } by A1;

then A3: x in the carrier of (W2 + W3) by Def1;

v in W1 by A2, Th3;

then x in { (v1 + v2) where v2, v1 is VECTOR of V : ( v1 in W1 & v2 in W2 ) } by A1;

then x in the carrier of (W1 + W2) by Def1;

then x in the carrier of (W1 + W2) /\ the carrier of (W2 + W3) by A3, XBOOLE_0:def 4;

hence x in the carrier of ((W1 + W2) /\ (W2 + W3)) by Def2; :: thesis: verum