let n be Nat; :: thesis: for K being Field
for V1 being VectSp of
for f being linear-transformation of V1,V1
for L being Scalar of holds f | (ker ((f + (L * (id V1))) |^ n)) is linear-transformation of ker ((f + (L * (id V1))) |^ n), ker ((f + (L * (id V1))) |^ n)
let K be Field; :: thesis: for V1 being VectSp of
for f being linear-transformation of V1,V1
for L being Scalar of holds f | (ker ((f + (L * (id V1))) |^ n)) is linear-transformation of ker ((f + (L * (id V1))) |^ n), ker ((f + (L * (id V1))) |^ n)
let V1 be VectSp of ; :: thesis: for f being linear-transformation of V1,V1
for L being Scalar of holds f | (ker ((f + (L * (id V1))) |^ n)) is linear-transformation of ker ((f + (L * (id V1))) |^ n), ker ((f + (L * (id V1))) |^ n)
let f be linear-transformation of V1,V1; :: thesis: for L being Scalar of holds f | (ker ((f + (L * (id V1))) |^ n)) is linear-transformation of ker ((f + (L * (id V1))) |^ n), ker ((f + (L * (id V1))) |^ n)
let L be Scalar of ; :: thesis: f | (ker ((f + (L * (id V1))) |^ n)) is linear-transformation of ker ((f + (L * (id V1))) |^ n), ker ((f + (L * (id V1))) |^ n)
set fid = f + (L * (id V1));
set KER = ker ((f + (L * (id V1))) |^ n);
reconsider fidK = (f + (L * (id V1))) | (ker ((f + (L * (id V1))) |^ n)) as linear-transformation of ker ((f + (L * (id V1))) |^ n), ker ((f + (L * (id V1))) |^ n) by Th28;
rng (f | (ker ((f + (L * (id V1))) |^ n))) c= the carrier of (ker ((f + (L * (id V1))) |^ n))

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (f | (ker ((f + (L * (id V1))) |^ n))) or y in the carrier of (ker ((f + (L * (id V1))) |^ n)) )
assume y in rng (f | (ker ((f + (L * (id V1))) |^ n))) ; :: thesis: y in the carrier of (ker ((f + (L * (id V1))) |^ n))
then consider x being set such that
A1: x in dom (f | (ker ((f + (L * (id V1))) |^ n))) and
A2: (f | (ker ((f + (L * (id V1))) |^ n))) . x = y by FUNCT_1:def 5;
A3: x in the carrier of (ker ((f + (L * (id V1))) |^ n)) by A1, FUNCT_2:def 1;
then A4: x in ker ((f + (L * (id V1))) |^ n) by STRUCT_0:def 5;
then x in V1 by VECTSP_4:17;
then reconsider v = x as Vector of by STRUCT_0:def 5;
A5: (f | (ker ((f + (L * (id V1))) |^ n))) . v = f . v by A1, FUNCT_1:70;
dom fidK = the carrier of (ker ((f + (L * (id V1))) |^ n)) by FUNCT_2:def 1;
then ( fidK . v = (f + (L * (id V1))) . v & fidK /. v = fidK . v ) by A3, FUNCT_1:70, PARTFUN1:def 8;
then A6: (f + (L * (id V1))) . v in ker ((f + (L * (id V1))) |^ n) by STRUCT_0:def 5;
(f + (L * (id V1))) . v = (f . v) + ((L * (id V1)) . v) by MATRLIN:def 5
.= (f . v) + (L * ((id V1) . v)) by MATRLIN:def 6
.= (f . v) + (L * v) by FUNCT_1:35 ;
then A7: ((f + (L * (id V1))) . v) + ((- L) * v) = (f . v) + ((L * v) + ((- L) * v)) by RLVECT_1:def 6
.= (f . v) + ((L + (- L)) * v) by VECTSP_1:def 26
.= (f . v) + ((0. K) * v) by VECTSP_1:63
.= (f . v) + (0. V1) by VECTSP_1:59
.= f . v by RLVECT_1:def 7 ;
(- L) * v in ker ((f + (L * (id V1))) |^ n) by A4, VECTSP_4:29;
then f . v in ker ((f + (L * (id V1))) |^ n) by A7, A6, VECTSP_4:28;
hence y in the carrier of (ker ((f + (L * (id V1))) |^ n)) by A2, A5, STRUCT_0:def 5; :: thesis: verum

end;

hence f | (ker ((f + (L * (id V1))) |^ n)) is linear-transformation of ker ((f + (L * (id V1))) |^ n), ker ((f + (L * (id V1))) |^ n) by Lm1, FUNCT_2:8; :: thesis: verum