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
  M1 is_anti-circular_about p & n>0 implies p = Line(M1,1)
proof
  assume that
A1: M1 is_anti-circular_about p and
A2: n>0;
A3: dom p=Seg len p by FINSEQ_1:def 3;
A4: width M1=n by MATRIX_0:24;
  then
A5: len p=n by A1;
A6: for k be Nat st k in dom p holds p.k = Line(M1,1).k
  proof
    let k be Nat;
    assume
A7: k in dom p;
    then
A8: 1<=k by A3,FINSEQ_1:1;
    n>=0+1 by A2,INT_1:7;
    then 1 in Seg n;
    then [1,k] in [:Seg n, Seg n:] by A3,A5,A7,ZFMISC_1:def 2;
    then
A9: [1,k] in Indices M1 by MATRIX_0:24;
A10: k<=n by A3,A5,A7,FINSEQ_1:1;
    Line(M1,1).k =M1*(1,k) by A4,A3,A5,A7,MATRIX_0:def 7
      .=p.((k-1 mod len p)+1) by A1,A8,A9
      .=p.((k-1 mod n)+1) by A1,A4
      .=p.(k-1+1) by A8,A10,Lm1;
    hence thesis;
  end;
  len Line(M1,1)=n by A4,MATRIX_0:def 7;
  then dom Line(M1,1)=dom p by A3,A5,FINSEQ_1:def 3;
  hence thesis by A6,FINSEQ_1:13;
end;
