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 implies a*M1 is anti-circular
proof
A1: Indices (a*M1)=[:Seg n, Seg n:] by MATRIX_0:24;
  assume M1 is anti-circular;
  then consider p being FinSequence of K such that
A2: len p=width M1 and
A3: M1 is_anti-circular_about p;
A4: width M1=n by MATRIX_0:24;
  then
A5: dom p=Seg n by A2,FINSEQ_1:def 3;
A6: dom (a*p)=Seg len (a*p) by FINSEQ_1:def 3;
A7: for i,j be Nat st [i,j] in Indices (a*M1)&i<=j holds (a*M1)*(i,j)=(a*p).
  ((j-i mod len (a*p))+1)
  proof
    let i,j be Nat;
    assume that
A8: [i,j] in Indices (a*M1) and
A9: i<=j;
A10: (j-i mod n)+1 in Seg n by A1,A8,Lm3;
    then
A11: (j-i mod len p)+1 in dom (a*p) by A2,A4,A6,MATRIXR1:16;
A12: [i,j] in Indices M1 by A1,A8,MATRIX_0:24;
    then (a*M1)*(i,j) =a*(M1*(i,j)) by MATRIX_3:def 5
      .=(a multfield).(M1*(i,j)) by FVSUM_1:49
      .=(a multfield).(p.((j-i mod len p)+1)) by A3,A9,A12
      .=(a multfield).(p/.((j-i mod len p)+1)) by A2,A4,A5,A10,PARTFUN1:def 6
      .=a*(p/.((j-i mod len p)+1)) by FVSUM_1:49
      .=(a*p)/.((j-i mod len p)+1) by A2,A4,A5,A10,POLYNOM1:def 1
      .=(a*p).((j-i mod len p)+1) by A11,PARTFUN1:def 6;
    hence thesis by MATRIXR1:16;
  end;
A13: width (a*M1)=n by MATRIX_0:24;
A14: len p =n by A2,MATRIX_0:24;
A15: len (a*p)=len p by MATRIXR1:16;
  for i,j be Nat st [i,j] in Indices (a*M1)&i>=j holds (a*M1)*(i,j)=(-(a*
  p)).((j-i mod len (a*p))+1)
  proof
    len (a*(-p))=len (-p) by MATRIXR1:16;
    then
A16: dom (a*(-p))=Seg len (-p) by FINSEQ_1:def 3
      .=dom (-p) by FINSEQ_1:def 3;
A17: a*p is Element of n-tuples_on the carrier of K by A15,A14,FINSEQ_2:92;
    let i,j be Nat;
    assume that
A18: [i,j] in Indices (a*M1) and
A19: i>=j;
A20: (j-i mod n)+1 in Seg n by A1,A18,Lm3;
A21: p is Element of n-tuples_on the carrier of K by A14,FINSEQ_2:92;
    then -p is Element of (len p)-tuples_on the carrier of K by A14,
FINSEQ_2:113;
    then len (-p)=len p by CARD_1:def 7;
    then
A22: dom (-p)=Seg n by A2,A4,FINSEQ_1:def 3;
A23: [i,j] in Indices M1 by A1,A18,MATRIX_0:24;
    then (a*M1)*(i,j)=a*(M1*(i,j)) by MATRIX_3:def 5
      .=(a multfield).(M1*(i,j)) by FVSUM_1:49
      .=(a multfield).((-p).((j-i mod len p)+1)) by A3,A19,A23
      .=(a multfield).((-p)/.((j-i mod len p)+1)) by A2,A4,A20,A22,
PARTFUN1:def 6
      .=a*((-p)/.((j-i mod len p)+1)) by FVSUM_1:49
      .=(a*(-p))/.((j-i mod len p)+1) by A2,A4,A20,A22,POLYNOM1:def 1
      .=(a*(-p)).((j-i mod len p)+1) by A2,A4,A20,A22,A16,PARTFUN1:def 6
      .=(a*((-1_K)*p)).((j-i mod len p)+1) by A21,FVSUM_1:59
      .=((a*(-1_K))*p).((j-i mod len p)+1) by A21,FVSUM_1:54
      .=((-a*1_K)*p).((j-i mod len p)+1) by VECTSP_1:8
      .=((-a)*p).((j-i mod len p)+1)
      .=((-1_K*a)*p).((j-i mod len p)+1)
      .=(((-1_K)*a)*p).((j-i mod len p)+1) by VECTSP_1:9
      .=((-1_K)*(a*p)).((j-i mod len p)+1) by A21,FVSUM_1:54
      .=(-(a*p)).((j-i mod len p)+1) by A17,FVSUM_1:59;
    hence thesis by MATRIXR1:16;
  end;
  then a*M1 is_anti-circular_about a*p by A13,A15,A14,A7;
  then consider q being FinSequence of K such that
A24: len q =width (a*M1) & (a*M1) is_anti-circular_about q;
  take q;
  thus thesis by A24;
end;
