let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for v being Element of V

for L1, L2 being Linear_Combination of V holds (L1 + L2) . v = (L1 . v) + (L2 . v)

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for v being Element of V

for L1, L2 being Linear_Combination of V holds (L1 + L2) . v = (L1 . v) + (L2 . v)

let v be Element of V; :: thesis: for L1, L2 being Linear_Combination of V holds (L1 + L2) . v = (L1 . v) + (L2 . v)

let L1, L2 be Linear_Combination of V; :: thesis: (L1 + L2) . v = (L1 . v) + (L2 . v)

( dom L1 = the carrier of V & dom L2 = the carrier of V ) by FUNCT_2:def 1;

then A1: ( L1 /. v = L1 . v & L2 /. v = L2 . v ) by PARTFUN1:def 6;

A2: dom (L1 + L2) = the carrier of V by FUNCT_2:def 1;

hence (L1 + L2) . v = (L1 + L2) /. v by PARTFUN1:def 6

.= (L1 . v) + (L2 . v) by A1, A2, VFUNCT_1:def 1 ;

:: thesis: verum

for v being Element of V

for L1, L2 being Linear_Combination of V holds (L1 + L2) . v = (L1 . v) + (L2 . v)

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for v being Element of V

for L1, L2 being Linear_Combination of V holds (L1 + L2) . v = (L1 . v) + (L2 . v)

let v be Element of V; :: thesis: for L1, L2 being Linear_Combination of V holds (L1 + L2) . v = (L1 . v) + (L2 . v)

let L1, L2 be Linear_Combination of V; :: thesis: (L1 + L2) . v = (L1 . v) + (L2 . v)

( dom L1 = the carrier of V & dom L2 = the carrier of V ) by FUNCT_2:def 1;

then A1: ( L1 /. v = L1 . v & L2 /. v = L2 . v ) by PARTFUN1:def 6;

A2: dom (L1 + L2) = the carrier of V by FUNCT_2:def 1;

hence (L1 + L2) . v = (L1 + L2) /. v by PARTFUN1:def 6

.= (L1 . v) + (L2 . v) by A1, A2, VFUNCT_1:def 1 ;

:: thesis: verum