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 Th7:
  M1 is line_circulant & M2 is line_circulant implies M1+M2 is line_circulant
proof
  assume that
A1: M1 is line_circulant and
A2: M2 is line_circulant;
  consider p being FinSequence of K such that
A3: len p=width M1 and
A4: M1 is_line_circulant_about p by A1;
A5: Indices M2=[:Seg n, Seg n:] by MATRIX_0:24;
A6: Indices (M1+M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A7: width M1=n by MATRIX_0:24;
  then
A8: dom p=Seg n by A3,FINSEQ_1:def 3;
  consider q being FinSequence of K such that
A9: len q=width M2 and
A10: M2 is_line_circulant_about q by A2;
A11: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A12: width M2=n by MATRIX_0:24;
  then dom q=Seg n by A9,FINSEQ_1:def 3;
  then
A13: dom (p+q)=dom p by A8,POLYNOM1:1;
  then
A14: len (p+q)=n by A8,FINSEQ_1:def 3;
A15: width (M1+M2)=n by MATRIX_0:24;
A16: dom (p+q)=Seg len (p+q) by FINSEQ_1:def 3;
  for i,j be Nat st [i,j] in Indices (M1+M2) holds (M1+M2)*(i,j)=(p+q).((
  j-i mod len (p+q))+1)
  proof
    let i,j be Nat;
    assume
A17: [i,j] in Indices (M1+M2);
    then
A18: (j-i mod len (p+q))+1 in dom (p+q) by A6,A8,A16,A13,Lm3;
    (M1+M2)*(i,j) =M1*(i,j) + M2*(i,j) by A11,A6,A17,MATRIX_3:def 3
      .=(the addF of K).(M1*(i,j),q.((j-i mod len q)+1)) by A10,A5,A6,A17
      .=(the addF of K).(p.((j-i mod len (p+q))+1),q.((j-i mod len (p+q))+1)
    ) by A4,A9,A11,A7,A12,A6,A14,A17
      .=(p+q).((j-i mod len (p+q))+1) by A18,FUNCOP_1:22;
    hence thesis;
  end;
  then M1+M2 is_line_circulant_about p+q by A15,A14;
  then consider r being FinSequence of K such that
A19: len r =width (M1+M2) & M1+M2 is_line_circulant_about r;
  take r;
  thus thesis by A19;
end;
