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 Th3:
  M1 is_symmetry_circulant_about p implies a*M1 is_symmetry_circulant_about a*p
  proof
    assume
A1: M1 is_symmetry_circulant_about p;
    then
A2: len p=width M1;
A3: Indices (a*M1)=[:Seg n, Seg n:] by MATRIX_0:24;
A4: width M1=n by MATRIX_0:24; then
A5: dom p=Seg n by A2,FINSEQ_1:def 3;
A6: dom (a*p)=Seg len (a*p) by FINSEQ_1:def 3;
A7:len (a*p)=len p by MATRIXR1:16;
A8: for i,j be Nat st [i,j] in Indices (a*M1)  & i+j<>len (a*p)+1 holds
   (a*M1)*(i,j)=(a*p).(i+j-1 mod len (a*p))
  proof
    let i,j be Nat;
    assume that
A9: [i,j] in Indices (a*M1) and
A10:i+j<>len (a*p)+1;
A11: i+j-1 mod n in Seg n by A3,A2,A4,A9,A10,A7,Lm4;
A12: [i,j] in Indices M1 by A3,A9,MATRIX_0:24;
    then (a*M1)*(i,j) =a*(M1*(i,j)) by MATRIX_3:def 5
      .=(a multfield).(M1*(i,j)) by FVSUM_1:49
      .=(a multfield).(p.(i+j-1 mod len p)) by A1,A12,A10,A7
      .=(a multfield).(p/.(i+j-1 mod len p)) by A2,A4,A5,A11,PARTFUN1:def 6
      .=a*(p/.(i+j-1 mod len p)) by FVSUM_1:49
      .=(a*p)/.(i+j-1 mod len p) by A2,A4,A5,A11,POLYNOM1:def 1
      .=(a*p).(i+j-1 mod len p) by A2,A4,A6,A7,A11,PARTFUN1:def 6;
    hence thesis by MATRIXR1:16;
  end;
A13:for i,j be Nat st [i,j] in Indices (a*M1)  & i+j=len (a*p)+1 holds
(a*M1)*(i,j)=(a*p).(len (a*p))
  proof
  let i,j be Nat;
    assume that
A14: [i,j] in Indices (a*M1) and
A15:i+j=len (a*p)+1;
       i in Seg n & j in Seg n by A3,A14,ZFMISC_1:87; then
       1<=i & 1<=j by FINSEQ_1:1;
       then 1+1<=i+j by XREAL_1:7; then
A16: len (a*p) +1-1 >=1+1-1 by A15,XREAL_1:9;
A17: len (a*p) in Seg len (a*p) by A16; then
A18:len p in dom (a*p) by A7,FINSEQ_1:def 3;
A19:len p in dom p by A7,A17,FINSEQ_1:def 3;
A20:[i,j] in Indices M1 by A3,A14,MATRIX_0:24;
      then (a*M1)*(i,j) =a*(M1*(i,j)) by MATRIX_3:def 5
      .=(a multfield).(M1*(i,j)) by FVSUM_1:49
      .=(a multfield).(p.(len p)) by A1,A20,A15,A7
      .=(a multfield).(p/.(len p)) by A19,PARTFUN1:def 6
      .=a*(p/.(len p)) by FVSUM_1:49
      .=(a*p)/.(len p) by A19,POLYNOM1:def 1
      .=(a*p).(len p) by A18,PARTFUN1:def 6;
     hence thesis by MATRIXR1:16;
   end;
A21: width (a*M1)=n & len (a*p)=len p by MATRIXR1:16,MATRIX_0:24;
    len p =n by A2,MATRIX_0:24;
    hence thesis by A21,A8,A13;
  end;
