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 Th52:
  M1 is anti-circular & M2 is anti-circular implies M1+M2 is anti-circular
proof
  assume that
A1: M1 is anti-circular and
A2: M2 is anti-circular;
  consider p being FinSequence of K such that
A3: len p=width M1 and
A4: M1 is_anti-circular_about p by A1;
A5: width M1=n by MATRIX_0:24;
  then
A6: dom p=Seg n by A3,FINSEQ_1:def 3;
  consider q being FinSequence of K such that
A7: len q=width M2 and
A8: M2 is_anti-circular_about q by A2;
A9: dom (p+q)=Seg len (p+q) by FINSEQ_1:def 3;
A10: width M2=n by MATRIX_0:24;
  then dom q=Seg n by A7,FINSEQ_1:def 3;
  then
A11: dom (p+q)=dom p by A6,POLYNOM1:1;
  then
A12: len (p+q)=n by A6,FINSEQ_1:def 3;
A13: p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:92;
  then -p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:113;
  then len (-p)=len p by CARD_1:def 7;
  then
A14: dom -p=Seg n by A3,A5,FINSEQ_1:def 3;
A15: Indices M2=[:Seg n, Seg n:] by MATRIX_0:24;
A16: Indices (M1+M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A17: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A18: q is Element of (len q)-tuples_on the carrier of K by FINSEQ_2:92;
  then -q is Element of (len q)-tuples_on the carrier of K by FINSEQ_2:113;
  then len (-q)=len q by CARD_1:def 7;
  then
A19: dom -q=Seg n by A7,A10,FINSEQ_1:def 3;
A20: for i,j be Nat st [i,j] in Indices (M1+M2)&i>=j holds (M1+M2)*(i,j)=(-(
  p+q)).((j-i mod len (p+q))+1)
  proof
    let i,j be Nat;
    assume that
A21: [i,j] in Indices (M1+M2) and
A22: i>=j;
    dom (-p+-q)=dom -p by A14,A19,POLYNOM1:1;
    then
A23: (j-i mod len (p+q))+1 in dom (-p+-q) by A16,A6,A9,A14,A11,A21,Lm3;
    (M1+M2)*(i,j) =M1*(i,j)+M2*(i,j) by A17,A16,A21,MATRIX_3:def 3
      .=(the addF of K).(M1*(i,j),(-q).((j-i mod len q)+1)) by A8,A15,A16,A21
,A22
      .=(the addF of K).((-p).((j-i mod len (p+q))+1),(-q).((j-i mod len (p+
    q))+1)) by A4,A7,A17,A5,A10,A16,A12,A21,A22
      .=(-p+-q).((j-i mod len (p+q))+1) by A23,FUNCOP_1:22
      .=(-(p+q)).((j-i mod len (p+q))+1) by A3,A7,A13,A18,A5,A10,FVSUM_1:31;
    hence thesis;
  end;
A24: width (M1+M2)=n by MATRIX_0:24;
  for i,j be Nat st [i,j] in Indices (M1+M2)&i<=j holds (M1+M2)*(i,j)=(p+q
  ).((j-i mod len (p+q))+1)
  proof
    let i,j be Nat;
    assume that
A25: [i,j] in Indices (M1+M2) and
A26: i<=j;
A27: (j-i mod len (p+q))+1 in dom (p+q) by A16,A6,A9,A11,A25,Lm3;
    (M1+M2)*(i,j) =M1*(i,j) + M2*(i,j) by A17,A16,A25,MATRIX_3:def 3
      .=(the addF of K).(M1*(i,j),q.((j-i mod len q)+1)) by A8,A15,A16,A25,A26

      .=(the addF of K).(p.((j-i mod len (p+q))+1),q.((j-i mod len (p+q))+1)
    ) by A4,A7,A17,A5,A10,A16,A12,A25,A26
      .=(p+q).((j-i mod len (p+q))+1) by A27,FUNCOP_1:22;
    hence thesis;
  end;
  then M1+M2 is_anti-circular_about p+q by A24,A12,A20;
  then consider r being FinSequence of K such that
A28: len r =width (M1+M2) & M1+M2 is_anti-circular_about r;
  take r;
  thus thesis by A28;
end;
