reserve i,j,k,n,l for Element of 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 Th6:
  M1 is line_circulant implies a*M1 is line_circulant
proof
A1: Indices (a*M1)=[:Seg n, Seg n:] by MATRIX_0:24;
  assume M1 is line_circulant;
  then consider p being FinSequence of K such that
A2: len p=width M1 and
A3: M1 is_line_circulant_about p;
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: for i,j be Nat st [i,j] in Indices (a*M1) holds (a*M1)*(i,j)=(a*p).((j-
  i mod len (a*p))+1)
  proof
    let i,j be Nat;
    assume
A8: [i,j] in Indices (a*M1);
    then
A9: (j-i mod n)+1 in Seg n by A1,Lm3;
    then
A10: (j-i mod len p)+1 in dom (a*p) by A2,A4,A6,MATRIXR1:16;
A11: [i,j] in Indices M1 by A1,A8,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.((j-i mod len p)+1)) by A3,A11
      .=(a multfield).(p/.((j-i mod len p)+1)) by A2,A4,A5,A9,PARTFUN1:def 6
      .=a*(p/.((j-i mod len p)+1)) by FVSUM_1:49
      .=(a*p)/.((j-i mod len p)+1) by A2,A4,A5,A9,POLYNOM1:def 1
      .=(a*p).((j-i mod len p)+1) by A10,PARTFUN1:def 6;
    hence thesis by MATRIXR1:16;
  end;
A12: width (a*M1)=n & len (a*p)=len p by MATRIXR1:16,MATRIX_0:24;
  len p =n by A2,MATRIX_0:24;
  then a*M1 is_line_circulant_about a*p by A12,A7;
  then consider q being FinSequence of K such that
A13: len q =width (a*M1) & (a*M1) is_line_circulant_about q;
  take q;
  thus thesis by A13;
end;
