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

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

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

set A = { (v + u) where u, v is VECTOR of V : ( v in W1 & u in W2 ) } ;

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

then reconsider v = x as Element of W1 ;

reconsider v = v as VECTOR of V by RLSUB_1:10;

A1: v = v + (0. V) ;

( v in W1 & 0. V in W2 ) by RLSUB_1:17, STRUCT_0:def 5;

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

hence x in the carrier of (W1 + W2) by Def1; :: thesis: verum

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

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

set A = { (v + u) where u, v is VECTOR of V : ( v in W1 & u in W2 ) } ;

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

then reconsider v = x as Element of W1 ;

reconsider v = v as VECTOR of V by RLSUB_1:10;

A1: v = v + (0. V) ;

( v in W1 & 0. V in W2 ) by RLSUB_1:17, STRUCT_0:def 5;

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

hence x in the carrier of (W1 + W2) by Def1; :: thesis: verum