let K be Field; :: thesis: for p being FinSequence of K st p is first-symmetry-of-circulant holds
- p is first-symmetry-of-circulant
let p be FinSequence of K; :: thesis: ( p is first-symmetry-of-circulant implies - p is first-symmetry-of-circulant )
set n = len p;
assume p is first-symmetry-of-circulant ; :: thesis: - p is first-symmetry-of-circulant
then consider M1 being Matrix of len p,K such that
A1: M1 is_symmetry_circulant_about p ;
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 A2: len (- p) = len p by CARD_1:def 7;
- M1 is_symmetry_circulant_about - p by A1, Th4;
then consider M2 being Matrix of len (- p),K such that
A3: M2 is_symmetry_circulant_about - p by A2;
take M2 ; :: according to MATRIX17:def 6 :: thesis: M2 is_symmetry_circulant_about - p
thus M2 is_symmetry_circulant_about - p by A3; :: thesis: verum