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 Th41:
  p is first-line-of-circulant implies a*p is first-line-of-circulant
proof
  set n=len p;
A1: len (a*p)=len p by MATRIXR1:16;
  assume p is first-line-of-circulant;
  then consider M1 being Matrix of n,K such that
A2: M1 is_line_circulant_about p;
A3: Indices (a*M1)=[:Seg n, Seg n:] by MATRIX_0:24;
A4: dom p=Seg n by FINSEQ_1:def 3;
A5: dom (a*p)=Seg len (a*p) by FINSEQ_1:def 3;
A6: 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
A7: [i,j] in Indices (a*M1);
    then
A8: (j-i mod n)+1 in Seg n by A3,Lm3;
    then
A9: (j-i mod len p)+1 in dom (a*p) by A5,MATRIXR1:16;
A10: [i,j] in Indices M1 by A3,A7,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 A2,A10
      .=(a multfield).(p/.((j-i mod len p)+1)) by A4,A8,PARTFUN1:def 6
      .=a*(p/.((j-i mod len p)+1)) by FVSUM_1:49
      .=(a*p)/.((j-i mod len p)+1) by A4,A8,POLYNOM1:def 1
      .=(a*p).((j-i mod len p)+1) by A9,PARTFUN1:def 6;
    hence thesis by MATRIXR1:16;
  end;
  width (a*M1)=n by MATRIX_0:24;
  then a*M1 is_line_circulant_about a*p by A1,A6;
  then consider M2 being Matrix of len (a*p),K such that
A11: M2 is_line_circulant_about a*p by A1;
  take M2;
  thus thesis by A11;
end;
