reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;

theorem Th13:
  for M be (Matrix of n,K),R be Permutation of Seg n st R=Rev
  idseq n & for i,j st i in Seg n & j in Seg n & i+j > n+1 holds M*(i,j) = 0.K
  holds M*R is lower_triangular Matrix of n,K
proof
  let M be (Matrix of n,K),R be Permutation of Seg n such that
A1: R=Rev idseq n and
A2: for i,j st i in Seg n & j in Seg n & i+j > n+1 holds M*(i,j)=0.K;
  set I=idseq n;
  set MR=M*R;
  now
    let i,j such that
A3: [i,j] in Indices MR and
A4: i<j;
    reconsider i9=i as Element of NAT by ORDINAL1:def 12;
A5: Indices MR=[:Seg n,Seg n:] by MATRIX_0:24;
    then
A6: i in Seg n by A3,ZFMISC_1:87;
    then i<= n by FINSEQ_1:1;
    then reconsider ni=n-i9+1 as Element of NAT by FINSEQ_5:1;
    n+1-i>n+1-j by A4,XREAL_1:15;
    then
A7: ni+j>(n+1-j)+j by XREAL_1:8;
A8: len I=n by CARD_1:def 7;
A9: Indices M=[:Seg n,Seg n:] by MATRIX_0:24;
A10: ni in Seg n by A6,FINSEQ_5:2;
    then
A11: I.ni=ni by FUNCT_1:17;
    j in Seg n by A3,A5,ZFMISC_1:87;
    then
A12: M*(ni,j)=0.K by A2,A7,A10;
    dom I=Seg len I by FINSEQ_1:def 3;
    then R.i=I.ni by A1,A6,A8,FINSEQ_5:58;
    hence MR*(i,j) = 0.K by A3,A5,A9,A11,A12,MATRIX11:def 4;
  end;
  hence thesis by MATRIX_1:def 9;
end;
