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;
reserve D for non empty set,
  t for FinSequence of D,
  A for Matrix of n,D;

theorem Th11:
  len p=len q & p is first-symmetry-of-circulant &
  q is first-symmetry-of-circulant implies SCirc(p+q) = SCirc(p)+SCirc(q)
proof
  set n = len p;
  assume that
A1: len p=len q and
A2: p is first-symmetry-of-circulant and
A3: q is first-symmetry-of-circulant;
A4: SCirc(q) is_symmetry_circulant_about q & Indices SCirc(p) =Indices SCirc(q)
  by A1,A3,Def7,MATRIX_0:26;
  p+q is first-symmetry-of-circulant by A1,A2,A3,Th10; then
A5: SCirc(p+q) is_symmetry_circulant_about (p+q) by Def7;
A6: dom p=Seg n by FINSEQ_1:def 3;
A7: SCirc(p) is_symmetry_circulant_about p by A2,Def7;
A8: dom (p+q)=Seg len (p+q) by FINSEQ_1:def 3;
A9: Indices SCirc(p) =[:Seg n, Seg n:] by MATRIX_0:24;
  dom q=Seg n by A1,FINSEQ_1:def 3;
  then dom (p+q)=dom p by A6,POLYNOM1:1; then
A10: len (p+q)=n by A6,FINSEQ_1:def 3; then
A11: Indices SCirc(p) =Indices SCirc(p+q) by MATRIX_0:26;
A12: for i,j be Nat holds [i,j] in Indices SCirc(p) implies SCirc(p+q)*(i,j)
  =SCirc(p)*(i,j)+SCirc(q)*(i,j)
  proof
    let i,j be Nat;
    assume
A13: [i,j] in Indices SCirc(p);
    now
      per cases;
      suppose
A14:   i+j<>len (p+q) +1;
A15: i+j-1 mod n in Seg n by A9,A13,A14,A10,Lm4;
         SCirc(p+q)*(i,j) =(p+q).(i+j-1 mod len (p+q))
         by A5,A11,A13,A14
         .=(the addF of K).(p.(i+j-1 mod len (p+q)),q.(i+j-1 mod len (p+q))
         ) by A8,A10,A15,FUNCOP_1:22
         .=(the addF of K).(SCirc(p)*(i,j),q.(i+j-1 mod len q))
         by A1,A10,A7,A13,A14
         .=SCirc(p)*(i,j) + SCirc(q)*(i,j) by A1,A4,A13,A14,A10;
         hence thesis;
      end;
      suppose
A16:  i+j=len (p+q) +1;
          i in Seg n & j in Seg n by A9,A13,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 A16,XREAL_1:9;
       then len (p+q) in Seg len (p+q);
       then
A17:  len (p+q) in dom (p+q) by FINSEQ_1:def 3;
         SCirc(p+q)*(i,j) =(p+q).(len (p+q)) by A5,A11,A13,A16
         .=(the addF of K).(p.(len (p+q)),q.(len (p+q))
         ) by A17,FUNCOP_1:22
         .=(the addF of K).(SCirc(p)*(i,j),q.(len q))
         by A1,A10,A7,A13,A16
         .=SCirc(p)*(i,j) + SCirc(q)*(i,j) by A4,A13,A16,A1,A10;
         hence thesis;
      end;
    end;
    hence thesis;
  end;
A18: len SCirc(p)= len p & width SCirc(p) = len p by MATRIX_0:24;
  len SCirc(p+q)= len p & width SCirc(p+q)=len p by A10,MATRIX_0:24;
  hence thesis by A18,A12,MATRIX_3:def 3;
end;
