let K be Field; :: thesis: for a being Element of K
for p being FinSequence of K st p is first-col-of-circulant holds
a * p is first-col-of-circulant
let a be Element of K; :: thesis: for p being FinSequence of K st p is first-col-of-circulant holds
a * p is first-col-of-circulant
let p be FinSequence of K; :: thesis: ( p is first-col-of-circulant implies a * p is first-col-of-circulant )
set n = len p;
A1: len (a * p) = len p by MATRIXR1:16;
assume p is first-col-of-circulant ; :: thesis: a * p is first-col-of-circulant
then consider M1 being Matrix of len p,K such that
A2: M1 is_col_circulant_about p ;
A3: Indices (a * M1) = [:(Seg (len p)),(Seg (len p)):] by MATRIX_0:24;
A4: dom p = Seg (len p) by FINSEQ_1:def 3;
A5: dom (a * p) = Seg (len (a * p)) by FINSEQ_1:def 3;
A6: for i, j being Nat st [i,j] in Indices (a * M1) holds
(a * M1) * (i,j) = (a * p) . (((i - j) mod (len (a * p))) + 1)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (a * M1) implies (a * M1) * (i,j) = (a * p) . (((i - j) mod (len (a * p))) + 1) )
assume A7: [i,j] in Indices (a * M1) ; :: thesis: (a * M1) * (i,j) = (a * p) . (((i - j) mod (len (a * p))) + 1)
then A8: ((i - j) mod (len p)) + 1 in Seg (len p) by A3, Lm3;
then A9: ((i - j) mod (len p)) + 1 in dom (a * p) by A5, MATRIXR1:16;
A10: [i,j] in Indices M1 by A3, A7, MATRIX_0:24;
then (a * M1) * (i,j) = a * (M1 * (i,j)) by MATRIX_3:def 5
.= (a multfield) . (M1 * (i,j)) by FVSUM_1:49
.= (a multfield) . (p . (((i - j) mod (len p)) + 1)) by A2, A10
.= (a multfield) . (p /. (((i - j) mod (len p)) + 1)) by A4, A8, PARTFUN1:def 6
.= a * (p /. (((i - j) mod (len p)) + 1)) by FVSUM_1:49
.= (a * p) /. (((i - j) mod (len p)) + 1) by A4, A8, POLYNOM1:def 1
.= (a * p) . (((i - j) mod (len p)) + 1) by A9, PARTFUN1:def 6 ;
hence (a * M1) * (i,j) = (a * p) . (((i - j) mod (len (a * p))) + 1) by MATRIXR1:16; :: thesis: verum

end;

len (a * M1) = len p by MATRIX_0:24;
then a * M1 is_col_circulant_about a * p by A1, A6;
then consider M2 being Matrix of len (a * p),K such that
A11: M2 is_col_circulant_about a * p by A1;
take M2 ; :: according to MATRIX16:def 6 :: thesis: M2 is_col_circulant_about a * p
thus M2 is_col_circulant_about a * p by A11; :: thesis: verum