let K be Field; :: thesis: for p, q being FinSequence of K st p is first-symmetry-of-circulant & q is first-symmetry-of-circulant & len p = len q holds
p + q is first-symmetry-of-circulant

let p, q be FinSequence of K; :: thesis: ( p is first-symmetry-of-circulant & q is first-symmetry-of-circulant & len p = len q implies p + q is first-symmetry-of-circulant )
set n = len p;
assume that
A1: p is first-symmetry-of-circulant and
A2: q is first-symmetry-of-circulant and
A3: len p = len q ; :: thesis:
consider M2 being Matrix of len p,K such that
A4: M2 is_symmetry_circulant_about q by A2, A3;
A5: dom p = Seg (len p) by FINSEQ_1:def 3;
dom q = Seg (len p) by ;
then dom (p + q) = dom p by ;
then A6: len (p + q) = len p by ;
consider M1 being Matrix of len p,K such that
A7: M1 is_symmetry_circulant_about p by A1;
width (M1 + M2) = len p by MATRIX_0:24;
then consider M3 being Matrix of len (p + q),K such that
len (p + q) = width M3 and
A8: M3 is_symmetry_circulant_about p + q by A6, A4, A7, Th5;
take M3 ; :: according to MATRIX17:def 6 :: thesis:
thus M3 is_symmetry_circulant_about p + q by A8; :: thesis: verum