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 Th13:
  p is first-symmetry-of-circulant implies SCirc(a*p) =a*(SCirc p)
proof
  set n=len p;
A1: len (a*p)=len p by MATRIXR1:16;
  assume
A2: p is first-symmetry-of-circulant;
  then a*p is first-symmetry-of-circulant by Th12;
  then
A3: SCirc(a*p) is_symmetry_circulant_about a*p by Def7;
A4: Indices SCirc(p) =[:Seg n, Seg n:] by MATRIX_0:24;
A5: SCirc(p) is_symmetry_circulant_about p by A2,Def7;
A6: for i,j be Nat st [i,j] in Indices SCirc(p) holds SCirc(a*p)*(i,j)=a*(
  SCirc(p)*(i,j))
  proof
    let i,j be Nat;
A7: dom (a*p)=Seg len (a*p) by FINSEQ_1:def 3;
    assume
A8: [i,j] in Indices SCirc(p);
    now
      per cases;
      suppose
A9:   i+j<>len p +1;
A10:   i+j-1 mod n in Seg n by A4,A8,A9,Lm4;
A11: dom p=Seg len p by FINSEQ_1:def 3;
         [i,j] in Indices SCirc(a*p) by A1,A8,MATRIX_0:26;
         then SCirc(a*p)*(i,j) =(a*p).(i+j-1 mod len (a*p)) by A1,A3,A9
         .=(a*p)/.(i+j-1 mod len p) by A1,A10,A7,PARTFUN1:def 6
         .=a*(p/.(i+j-1 mod len p)) by A10,A11,POLYNOM1:def 1
         .=(a multfield).(p/.(i+j-1 mod len p)) by FVSUM_1:49
         .=(a multfield).(p.(i+j-1 mod len p)) by A10,A11,PARTFUN1:def 6
         .=(a multfield).(SCirc(p)*(i,j)) by A5,A8,A9
         .=a*(SCirc(p)*(i,j)) by FVSUM_1:49;
         hence thesis;
       end;
       suppose
A12:   i+j=len p +1;
A13:[i,j] in Indices SCirc(a*p) by A1,A8,MATRIX_0:26;
         i in Seg n & j in Seg n by A4,A8,ZFMISC_1:87; then
        1<=i & 1<=j by FINSEQ_1:1;
        then 1+1<=i+j by XREAL_1:7;
        then len p +1-1 >=1+1-1 by A12,XREAL_1:9;
        then
A14:    len p in Seg len p; then
A15: len p in dom (a*p) by A1,FINSEQ_1:def 3;
A16:  len p in dom p by A14,FINSEQ_1:def 3;
         SCirc(a*p)*(i,j) =(a*p).(len (a*p)) by A1,A3,A12,A13
         .=(a*p)/.(len p) by A1,A15,PARTFUN1:def 6
         .=a*(p/.(len p)) by A16,POLYNOM1:def 1
         .=(a multfield).(p/.(len p)) by FVSUM_1:49
         .=(a multfield).(p.(len p)) by A16,PARTFUN1:def 6
         .=(a multfield).(SCirc(p)*(i,j)) by A5,A8,A12
         .=a*(SCirc(p)*(i,j)) by FVSUM_1:49;
         hence thesis;
       end;
      end;
      hence thesis;
    end;
A17: len SCirc(p)= len p & width SCirc(p) = len p by MATRIX_0:24;
  len SCirc(a*p)= len p & width SCirc(a*p) = len p by A1,MATRIX_0:24;
  hence thesis by A17,A6,MATRIX_3:def 5;
end;
