let V be RealLinearSpace; :: thesis: for L1, L2 being Linear_Combination of V holds Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2)

let L1, L2 be Linear_Combination of V; :: thesis: Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2)

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier (L1 + L2) or x in (Carrier L1) \/ (Carrier L2) )

assume x in Carrier (L1 + L2) ; :: thesis: x in (Carrier L1) \/ (Carrier L2)

then consider u being VECTOR of V such that

A1: x = u and

A2: (L1 + L2) . u <> 0 ;

(L1 + L2) . u = (L1 . u) + (L2 . u) by Def10;

then ( L1 . u <> 0 or L2 . u <> 0 ) by A2;

then ( x in { v1 where v1 is VECTOR of V : L1 . v1 <> 0 } or x in { v2 where v2 is VECTOR of V : L2 . v2 <> 0 } ) by A1;

hence x in (Carrier L1) \/ (Carrier L2) by XBOOLE_0:def 3; :: thesis: verum

let L1, L2 be Linear_Combination of V; :: thesis: Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2)

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier (L1 + L2) or x in (Carrier L1) \/ (Carrier L2) )

assume x in Carrier (L1 + L2) ; :: thesis: x in (Carrier L1) \/ (Carrier L2)

then consider u being VECTOR of V such that

A1: x = u and

A2: (L1 + L2) . u <> 0 ;

(L1 + L2) . u = (L1 . u) + (L2 . u) by Def10;

then ( L1 . u <> 0 or L2 . u <> 0 ) by A2;

then ( x in { v1 where v1 is VECTOR of V : L1 . v1 <> 0 } or x in { v2 where v2 is VECTOR of V : L2 . v2 <> 0 } ) by A1;

hence x in (Carrier L1) \/ (Carrier L2) by XBOOLE_0:def 3; :: thesis: verum