let F be Field; :: thesis: for V, W being VectSp of F
for T being linear-transformation of V,W
for l being Linear_Combination of V
for w being Element of W st w in Carrier (T @ l) holds
T " {w} meets Carrier l

let V, W be VectSp of F; :: thesis: for T being linear-transformation of V,W
for l being Linear_Combination of V
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 l being Linear_Combination of V
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 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:
then A1: (T @ l) . w <> 0. F by VECTSP_6:2;
assume A2: T " {w} misses Carrier l ; :: thesis: contradiction
per cases ( T " {w} = {} or T " {w} <> {} ) ;
suppose T " {w} = {} ; :: thesis: contradiction
then Sum (l .: (T " {w})) = Sum ({} F)
.= 0. F by RLVECT_2:8 ;
hence contradiction by A1, Def5; :: thesis: verum
end;
suppose T " {w} <> {} ; :: thesis: contradiction
then l .: (T " {w}) = {(0. F)} by ;
then Sum (l .: (T " {w})) = 0. F by RLVECT_2:9;
hence contradiction by A1, Def5; :: thesis: verum
end;
end;