let K be Field; :: thesis: for V being VectSp of K
for KL1, KL2 being Linear_Combination of V
for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Sum KL1 = Sum KL2 holds
KL1 = KL2
let V be VectSp of K; :: thesis: for KL1, KL2 being Linear_Combination of V
for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Sum KL1 = Sum KL2 holds
KL1 = KL2
let KL1, KL2 be Linear_Combination of V; :: thesis: for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Sum KL1 = Sum KL2 holds
KL1 = KL2
let X be Subset of V; :: thesis: ( X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Sum KL1 = Sum KL2 implies KL1 = KL2 )
assume that
A1: X is linearly-independent and
A2: ( Carrier KL1 c= X & Carrier KL2 c= X ) and
A3: Sum KL1 = Sum KL2 ; :: thesis: KL1 = KL2
(Sum KL1) - (Sum KL2) = 0. V by A3, VECTSP_1:19;
then A4: ( KL1 - KL2 is Linear_Combination of Carrier (KL1 - KL2) & Sum (KL1 - KL2) = 0. V ) by VECTSP_6:7, VECTSP_6:47;
A5: Carrier (KL1 - KL2) c= (Carrier KL1) \/ (Carrier KL2) by VECTSP_6:41;
(Carrier KL1) \/ (Carrier KL2) c= X by A2, XBOOLE_1:8;
then A6: Carrier (KL1 - KL2) is linearly-independent by A1, A5, VECTSP_7:1, XBOOLE_1:1;

now :: thesis: for v being Vector of V holds KL1 . v = KL2 . v

let v be Vector of V; :: thesis: KL1 . v = KL2 . v
not v in Carrier (KL1 - KL2) by A6, A4, VECTSP_7:def 1;
then (KL1 - KL2) . v = 0. K by VECTSP_6:2;
then (KL1 . v) - (KL2 . v) = 0. K by VECTSP_6:40;
hence KL1 . v = KL2 . v by RLVECT_1:21; :: thesis: verum

end;

hence KL1 = KL2 by VECTSP_6:def 7; :: thesis: verum