reserve i,j,k,n,l for Nat,
  K for Field,
  a,b,c for Element of K,
  p,q for FinSequence of K,
  M1,M2,M3 for Matrix of n,K;

theorem Th5:
  M1 is_symmetry_circulant_about p & M2 is_symmetry_circulant_about q
  implies M1+M2 is_symmetry_circulant_about p+q
  proof
    assume that
A1: M1 is_symmetry_circulant_about p;
A2: len p=width M1 by A1;
A3: Indices M2=[:Seg n, Seg n:] by MATRIX_0:24;
A4: Indices (M1+M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A5: width M1=n by MATRIX_0:24; then
A6: dom p=Seg n by A2,FINSEQ_1:def 3;
    assume
A7: M2 is_symmetry_circulant_about q;
    then
A8: len q=width M2;
A9: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A10:  n in NAT by ORDINAL1:def 12;
A11: width M2=n by MATRIX_0:24;
  then dom q=Seg n by A8,FINSEQ_1:def 3;
  then
A12:dom (p+q)=dom p by A6,POLYNOM1:1;
  then
A13: len (p+q)=n by A6,A10,FINSEQ_1:def 3;
A14: width (M1+M2)=n by MATRIX_0:24;
A15: dom (p+q)=Seg len (p+q) by FINSEQ_1:def 3;
A16:for i,j be Nat st [i,j] in Indices (M1+M2) & i+j<>len (p+q) +1 holds
(M1+M2)*(i,j)=(p+q).(i+j-1 mod len (p+q))
  proof
    let i,j be Nat;
    assume that
A17: [i,j] in Indices (M1+M2) and
A18:i+j<>len (p+q) +1;
A19: ((i+j-1) mod len (p+q)) in dom (p+q)
        by A4,A15,A18,A17,A12,A6,Lm4;
    (M1+M2)*(i,j) =M1*(i,j) + M2*(i,j) by A9,A4,A17,MATRIX_3:def 3
      .=(the addF of K).(M1*(i,j),q.(i+j-1 mod len (p+q)))
      by A7,A3,A4,A11,A17,A18,A13
      .=(the addF of K).(p.(i+j-1 mod len (p+q)),q.((i+j-1 mod len (p+q))))
      by A1,A5,A9,A4,A13,A17,A18
      .=(p+q).(i+j-1 mod len (p+q)) by A19,FUNCOP_1:22;
    hence thesis;
  end;
  for i,j be Nat st [i,j] in Indices (M1+M2) & i+j=len (p+q) +1 holds
(M1+M2)*(i,j)=(p+q).(len (p+q))
  proof
   let i,j be Nat;
    assume that
A20: [i,j] in Indices (M1+M2) and
A21:i+j=len (p+q) +1;
       i in Seg n & j in Seg n by A4,A20,ZFMISC_1:87;
       then
       1<=i & 1<=j by FINSEQ_1:1;
       then 1+1<=i+j by XREAL_1:7;
       then len (p+q) +1-1 >=1+1-1 by A21,XREAL_1:9;
      then len (p+q) in Seg len (p+q);
       then
A22: len (p+q) in dom (p+q) by FINSEQ_1:def 3;
   (M1+M2)*(i,j) =M1*(i,j) + M2*(i,j) by A9,A4,A20,MATRIX_3:def 3
      .=(the addF of K).(M1*(i,j),q.(len (p+q))) by A7,A11,A13,A3,A4,A20,A21

      .=(the addF of K).(p.(len (p+q)),q.(len (p+q)))
      by A1,A9,A5,A4,A13,A20,A21
      .=(p+q).(len (p+q)) by A22,FUNCOP_1:22;
    hence thesis;
  end;
  hence thesis by A14,A13,A16;
 end;
