let V be free Z_Module; :: thesis: for KL1, KL2, KL3 being Linear_Combination of V
for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Carrier KL3 c= X & Sum KL1 = (Sum KL2) + (Sum KL3) holds
KL1 = KL2 + KL3
let KL1, KL2, KL3 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 & Carrier KL3 c= X & Sum KL1 = (Sum KL2) + (Sum KL3) holds
KL1 = KL2 + KL3
let X be Subset of V; :: thesis: ( X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Carrier KL3 c= X & Sum KL1 = (Sum KL2) + (Sum KL3) implies KL1 = KL2 + KL3 )
assume that
A1: ( X is linearly-independent & Carrier KL1 c= X ) and
A2: ( Carrier KL2 c= X & Carrier KL3 c= X ) and
A3: Sum KL1 = (Sum KL2) + (Sum KL3) ; :: thesis: KL1 = KL2 + KL3
( Carrier (KL2 + KL3) c= (Carrier KL2) \/ (Carrier KL3) & (Carrier KL2) \/ (Carrier KL3) c= X ) by A2, ZMODUL02:26, XBOOLE_1:8;
then A4: Carrier (KL2 + KL3) c= X ;
Sum KL1 = Sum (KL2 + KL3) by A3, ZMODUL02:52;
hence KL1 = KL2 + KL3 by A1, A4, Th5; :: thesis: verum