let K be Field; :: thesis: for p being FinSequence of K st p is first-col-of-circulant holds
- p is first-col-of-circulant
let p be FinSequence of K; :: thesis: ( p is first-col-of-circulant implies - p is first-col-of-circulant )
set n = len p;
assume p is first-col-of-circulant ; :: thesis: - p is first-col-of-circulant
then consider M1 being Matrix of len p,K such that
A1: M1 is_col_circulant_about p ;
set M2 = - M1;
A2: Indices M1 = [:(Seg (len p)),(Seg (len p)):] by MATRIX_0:24;
A3: Indices (- M1) = [:(Seg (len p)),(Seg (len p)):] by MATRIX_0:24;
p is Element of (len p) -tuples_on the carrier of K by FINSEQ_2:92;
then A4: - p is Element of (len p) -tuples_on the carrier of K by FINSEQ_2:113;
then A5: len (- p) = len p by CARD_1:def 7;
A6: dom p = Seg (len p) by FINSEQ_1:def 3;
A7: for i, j being Nat st [i,j] in Indices (- M1) holds
(- M1) * (i,j) = (- p) . (((i - j) mod (len (- p))) + 1)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (- M1) implies (- M1) * (i,j) = (- p) . (((i - j) mod (len (- p))) + 1) )
assume A8: [i,j] in Indices (- M1) ; :: thesis: (- M1) * (i,j) = (- p) . (((i - j) mod (len (- p))) + 1)
then A9: ((i - j) mod (len p)) + 1 in Seg (len p) by A3, Lm3;
(- M1) * (i,j) = - (M1 * (i,j)) by A2, A3, A8, MATRIX_3:def 2
.= (comp K) . (M1 * (i,j)) by VECTSP_1:def 13
.= (comp K) . (p . (((i - j) mod (len p)) + 1)) by A1, A2, A3, A8
.= (- p) . (((i - j) mod (len p)) + 1) by A6, A9, FUNCT_1:13 ;
hence (- M1) * (i,j) = (- p) . (((i - j) mod (len (- p))) + 1) by A4, CARD_1:def 7; :: thesis: verum

end;

len (- M1) = len p by MATRIX_0:24;
then - M1 is_col_circulant_about - p by A5, A7;
then consider M2 being Matrix of len (- p),K such that
A10: M2 is_col_circulant_about - p by A5;
take M2 ; :: according to MATRIX16:def 6 :: thesis: M2 is_col_circulant_about - p
thus M2 is_col_circulant_about - p by A10; :: thesis: verum