let V be free Z_Module; :: thesis: for a being Element of INT.Ring
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 & a <> 0. INT.Ring & Sum KL1 = a * (Sum KL2) holds
KL1 = a * KL2
let a be Element of INT.Ring; :: 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 & a <> 0. INT.Ring & Sum KL1 = a * (Sum KL2) holds
KL1 = a * 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 & a <> 0. INT.Ring & Sum KL1 = a * (Sum KL2) holds
KL1 = a * KL2
let X be Subset of V; :: thesis: ( X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & a <> 0. INT.Ring & Sum KL1 = a * (Sum KL2) implies KL1 = a * KL2 )
assume that
A1: ( X is linearly-independent & Carrier KL1 c= X ) and
A2: ( Carrier KL2 c= X & a <> 0. INT.Ring & Sum KL1 = a * (Sum KL2) ) ; :: thesis: KL1 = a * KL2
( Carrier (a * KL2) c= X & Sum KL1 = Sum (a * KL2) ) by A2, ZMODUL02:29, ZMODUL02:53;
hence KL1 = a * KL2 by A1, Th5; :: thesis: verum