let R be Ring; :: thesis: for V, W being LeftMod of R

for l being Linear_Combination of V

for T being linear-transformation of V,W

for w being Element of W st w in Carrier (T @* l) holds

T " {w} meets Carrier l

let V, W be LeftMod of R; :: thesis: for l being Linear_Combination of V

for T being linear-transformation of V,W

for w being Element of W st w in Carrier (T @* l) holds

T " {w} meets Carrier l

let l be Linear_Combination of V; :: thesis: for T being linear-transformation of V,W

for w being Element of W st w in Carrier (T @* l) holds

T " {w} meets Carrier l

let T be linear-transformation of V,W; :: thesis: for w being Element of W st w in Carrier (T @* l) holds

T " {w} meets Carrier l

let w be Element of W; :: thesis: ( w in Carrier (T @* l) implies T " {w} meets Carrier l )

assume w in Carrier (T @* l) ; :: thesis: T " {w} meets Carrier l

then A1: (T @* l) . w <> 0. R by ZMODUL02:8;

assume T " {w} misses Carrier l ; :: thesis: contradiction

then lCFST (l,T,w) = <*> the carrier of R ;

then Sum (lCFST (l,T,w)) = 0. R by RLVECT_1:43;

hence contradiction by A1, LDef5; :: thesis: verum

for l being Linear_Combination of V

for T being linear-transformation of V,W

for w being Element of W st w in Carrier (T @* l) holds

T " {w} meets Carrier l

let V, W be LeftMod of R; :: thesis: for l being Linear_Combination of V

for T being linear-transformation of V,W

for w being Element of W st w in Carrier (T @* l) holds

T " {w} meets Carrier l

let l be Linear_Combination of V; :: thesis: for T being linear-transformation of V,W

for w being Element of W st w in Carrier (T @* l) holds

T " {w} meets Carrier l

let T be linear-transformation of V,W; :: thesis: for w being Element of W st w in Carrier (T @* l) holds

T " {w} meets Carrier l

let w be Element of W; :: thesis: ( w in Carrier (T @* l) implies T " {w} meets Carrier l )

assume w in Carrier (T @* l) ; :: thesis: T " {w} meets Carrier l

then A1: (T @* l) . w <> 0. R by ZMODUL02:8;

assume T " {w} misses Carrier l ; :: thesis: contradiction

then lCFST (l,T,w) = <*> the carrier of R ;

then Sum (lCFST (l,T,w)) = 0. R by RLVECT_1:43;

hence contradiction by A1, LDef5; :: thesis: verum