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 Th61:
  p is first-line-of-anti-circular & q is
first-line-of-anti-circular & len p = len q implies ACirc(p+q) = ACirc(p)+ACirc
  (q)
proof
  set n = len p;
  assume that
A1: p is first-line-of-anti-circular and
A2: q is first-line-of-anti-circular and
A3: len p = len q;
A4: dom p=Seg n by FINSEQ_1:def 3;
  dom q=Seg n by A3,FINSEQ_1:def 3;
  then
A5: dom (p+q)=dom p by A4,POLYNOM1:1;
  then
A6: len (p+q)=n by A4,FINSEQ_1:def 3;
  then
A7: Indices ACirc(p) =Indices ACirc(p+q) by MATRIX_0:26;
A8: Indices ACirc(p) =Indices ACirc(q) by A3,MATRIX_0:26;
A9: dom (p+q)=Seg len (p+q) by FINSEQ_1:def 3;
A10: 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
A11: len (-p)=len p by CARD_1:def 7;
A12: Indices ACirc(p)=[:Seg n, Seg n:] by MATRIX_0:24;
  p+q is first-line-of-anti-circular by A1,A2,A3,Th60;
  then
A13: ACirc(p+q) is_anti-circular_about (p+q) by Def12;
A14: ACirc(q) is_anti-circular_about q by A2,Def12;
A15: 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
A16: len (-q)=len q by CARD_1:def 7;
A17: ACirc(p) is_anti-circular_about p by A1,Def12;
A18: Indices ACirc(q)=[:Seg n, Seg n:] by A3,MATRIX_0:24;
A19: for i,j be Nat holds [i,j] in Indices ACirc(p) implies ACirc(p+q)*(i,j)
  =ACirc(p)*(i,j)+ACirc(q)*(i,j)
  proof
    let i,j be Nat;
    assume
A20: [i,j] in Indices ACirc(p);
    then
A21: (j-i mod n)+1 in Seg n by A12,Lm3;
    now
      per cases;
      case
A22:    i<=j;
        then ACirc(p+q)*(i,j) =(p+q).((j-i mod len (p+q))+1) by A13,A7,A20
          .=(the addF of K).(p.((j-i mod len (p+q))+1),q.((j-i mod len (p+q)
        )+1)) by A9,A6,A21,FUNCOP_1:22
          .=(the addF of K).(ACirc(p)*(i,j),q.((j-i mod len q)+1)) by A3,A6,A17
,A20,A22
          .=ACirc(p)*(i,j) + ACirc(q)*(i,j) by A14,A8,A20,A22;
        hence thesis;
      end;
      case
A23:    i>=j;
A24:    dom (-p)=Seg len p by A11,FINSEQ_1:def 3;
        dom (-q)=Seg len q by A16,FINSEQ_1:def 3;
        then dom p=Seg n & dom (-p+-q)=dom -p by A3,A24,FINSEQ_1:def 3
,POLYNOM1:1;
        then
A25:    (j-i mod len (p+q))+1 in dom (-p+-q) by A9,A5,A12,A20,A24,Lm3;
        ACirc(p+q)*(i,j) =(-(p+q)).((j-i mod len (p+q))+1) by A13,A7,A20,A23

          .=(-p+-q).((j-i mod len (p+q))+1) by A3,A10,A15,FVSUM_1:31
          .=(the addF of K).((-p).((j-i mod len (p+q))+1),(-q).((j-i mod len
        (p+q))+1)) by A25,FUNCOP_1:22
          .=(the addF of K).(ACirc(p)*(i,j),(-q).((j-i mod len q)+1)) by A3,A6
,A17,A20,A23
          .=ACirc(p)*(i,j) + ACirc(q)*(i,j) by A14,A12,A18,A20,A23;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
A26: len ACirc(p)= len p & width ACirc(p) = len p by MATRIX_0:24;
  len ACirc(p+q)= len p & width ACirc(p+q) = len p by A6,MATRIX_0:24;
  hence thesis by A26,A19,MATRIX_3:def 3;
end;
