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 Th42:
  p is first-line-of-circulant implies LCirc(a*p) =a*(LCirc p)
proof
  set n=len p;
A1: len (a*p)=len p by MATRIXR1:16;
  assume
A2: p is first-line-of-circulant;
  then a*p is first-line-of-circulant by Th41;
  then
A3: LCirc(a*p) is_line_circulant_about a*p by Def7;
A4: Indices LCirc(p) =[:Seg n, Seg n:] by MATRIX_0:24;
A5: LCirc(p) is_line_circulant_about p by A2,Def7;
A6: for i,j be Nat st [i,j] in Indices LCirc(p) holds LCirc(a*p)*(i,j)=a*(
  LCirc(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 LCirc(p);
    then
A9: (j-i mod n)+1 in Seg n by A4,Lm3;
A10: dom p=Seg len p by FINSEQ_1:def 3;
    [i,j] in Indices LCirc(a*p) by A1,A8,MATRIX_0:26;
    then LCirc(a*p)*(i,j) =(a*p).((j-i mod len (a*p))+1) by A3
      .=(a*p)/.((j-i mod len p)+1) by A1,A9,A7,PARTFUN1:def 6
      .=a*(p/.((j-i mod len p)+1)) by A9,A10,POLYNOM1:def 1
      .=(a multfield).(p/.((j-i mod len p)+1)) by FVSUM_1:49
      .=(a multfield).(p.((j-i mod len p)+1)) by A9,A10,PARTFUN1:def 6
      .=(a multfield).(LCirc(p)*(i,j)) by A5,A8
      .=a*(LCirc(p)*(i,j)) by FVSUM_1:49;
    hence thesis;
  end;
A11: len LCirc(p)= len p & width LCirc(p) = len p by MATRIX_0:24;
  len LCirc(a*p)= len p & width LCirc(a*p) = len p by A1,MATRIX_0:24;
  hence thesis by A11,A6,MATRIX_3:def 5;
end;
