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 Th60:
  p is first-line-of-anti-circular & q is
  first-line-of-anti-circular & len p = len q implies p+q is
  first-line-of-anti-circular
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;
  consider M2 being Matrix of n,K such that
A4: M2 is_anti-circular_about q by A2,A3;
A5: width M2=n by MATRIX_0:24;
A6: dom p=Seg n by FINSEQ_1:def 3;
  len q=width M2 by A4;
  then dom q=Seg n by A5,FINSEQ_1:def 3;
  then
A7: dom (p+q)=dom p by A6,POLYNOM1:1;
  then
A8: len (p+q)=n by A6,FINSEQ_1:def 3;
  consider M1 being Matrix of n,K such that
A9: M1 is_anti-circular_about p by A1;
A10: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
  set M3=M1+M2;
A11: 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
A12: len (-q)=len q by CARD_1:def 7;
A13: Indices M2=[:Seg n, Seg n:] by MATRIX_0:24;
A14: Indices (M1+M2) = [:Seg n, Seg n:] by MATRIX_0:24;
A15: dom (p+q)=Seg len (p+q) by FINSEQ_1:def 3;
A16: 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
A17: [i,j] in Indices (M1+M2) and
A18: i<=j;
A19: (j-i mod len (p+q))+1 in dom (p+q) by A14,A6,A15,A7,A17,Lm3;
    (M1+M2)*(i,j) =M1*(i,j) + M2*(i,j) by A10,A14,A17,MATRIX_3:def 3
      .=(the addF of K).(M1*(i,j),q.((j-i mod len q)+1)) by A4,A13,A14,A17,A18

      .=(the addF of K).(p.((j-i mod len (p+q))+1),q.((j-i mod len (p+q))+1)
    ) by A3,A9,A10,A14,A8,A17,A18
      .=(p+q).((j-i mod len (p+q))+1) by A19,FUNCOP_1:22;
    hence thesis;
  end;
A20: 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
A21: len (-p)=len p by CARD_1:def 7;
  then
A22: dom -p=Seg n by FINSEQ_1:def 3;
A23: 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
A24: [i,j] in Indices (M1+M2) and
A25: i>=j;
    dom (-p)=Seg len p & dom (-q)=Seg len q by A21,A12,FINSEQ_1:def 3;
    then dom (-p+-q)=dom -p by A3,POLYNOM1:1;
    then
A26: (j-i mod len (p+q))+1 in dom (-p+-q) by A14,A6,A15,A22,A7,A24,Lm3;
    (M1+M2)*(i,j) =M1*(i,j) + M2*(i,j) by A10,A14,A24,MATRIX_3:def 3
      .=(the addF of K).(M1*(i,j),(-q).((j-i mod len q)+1)) by A4,A13,A14,A24
,A25
      .=(the addF of K).((-p).((j-i mod len p)+1),(-q).((j-i mod len q)+1))
    by A9,A10,A14,A24,A25
      .=(-p+-q).((j-i mod len (p+q))+1) by A3,A8,A26,FUNCOP_1:22
      .=(-(p+q)).((j-i mod len (p+q))+1) by A3,A20,A11,FVSUM_1:31;
    hence thesis;
  end;
  width (M1+M2)=n by MATRIX_0:24;
  then len (p+q)=width (M1+M2) by A6,A7,FINSEQ_1:def 3;
  then len (M1+M2)=n & M3 is_anti-circular_about (p+q) by A16,A23,
MATRIX_0:24;
  then consider M3 being Matrix of len (p+q),K such that
  len (p+q)=len M3 and
A27: M3 is_anti-circular_about p+q by A8;
  take M3;
  thus thesis by A27;
end;
