let K be Field; :: thesis: for p being FinSequence of K st p is first-line-of-circulant holds
LCirc (- p) = - (LCirc p)
let p be FinSequence of K; :: thesis: ( p is first-line-of-circulant implies LCirc (- p) = - (LCirc p) )
set n = len p;
A1: ( len (LCirc p) = len p & width (LCirc p) = len p ) by MATRIX_0:24;
A2: Indices (LCirc p) = [:(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 - p is Element of (len p) -tuples_on the carrier of K by FINSEQ_2:113;
then A3: len (- p) = len p by CARD_1:def 7;
assume A4: p is first-line-of-circulant ; :: thesis: LCirc (- p) = - (LCirc p)
then - p is first-line-of-circulant by Th31;
then A5: LCirc (- p) is_line_circulant_about - p by Def7;
A6: LCirc p is_line_circulant_about p by A4, Def7;
A7: for i, j being Nat st [i,j] in Indices (LCirc p) holds
(LCirc (- p)) * (i,j) = - ((LCirc p) * (i,j))

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (LCirc p) implies (LCirc (- p)) * (i,j) = - ((LCirc p) * (i,j)) )
assume A8: [i,j] in Indices (LCirc p) ; :: thesis: (LCirc (- p)) * (i,j) = - ((LCirc p) * (i,j))
then ((j - i) mod (len p)) + 1 in Seg (len p) by A2, Lm3;
then A9: ((j - i) mod (len p)) + 1 in dom p by FINSEQ_1:def 3;
[i,j] in Indices (LCirc (- p)) by A3, A8, MATRIX_0:26;
then (LCirc (- p)) * (i,j) = (- p) . (((j - i) mod (len (- p))) + 1) by A5
.= (comp K) . (p . (((j - i) mod (len p)) + 1)) by A3, A9, FUNCT_1:13
.= (comp K) . ((LCirc p) * (i,j)) by A6, A8
.= - ((LCirc p) * (i,j)) by VECTSP_1:def 13 ;
hence (LCirc (- p)) * (i,j) = - ((LCirc p) * (i,j)) ; :: thesis: verum

end;

( len (LCirc (- p)) = len p & width (LCirc (- p)) = len p ) by A3, MATRIX_0:24;
hence LCirc (- p) = - (LCirc p) by A1, A7, MATRIX_3:def 2; :: thesis: verum