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;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;

theorem Th100:
  for M be diagonal Matrix of K for P st [:P,P:] c= Indices M
  holds Segm(M,P,P) is diagonal
proof
  let M be diagonal Matrix of K;
  let P such that
A1: [:P,P:] c= Indices M;
  set S=Segm(M,P,P);
  set SP=Sgm P;
  let i,j be Nat such that
A2: i in Seg card P and
A3: j in Seg card P and
A4: i <> j;
A6: SP is one-to-one by FINSEQ_3:92;
  [i,j] in [:Seg card P,Seg card P:] by A2,A3,ZFMISC_1:87;
  then
A7: [i,j] in Indices S by MATRIX_0:24;
  dom SP=Seg card P by FINSEQ_3:40;
  then
A8: SP.i<>SP.j by A2,A3,A4,A6;
  rng SP=P by FINSEQ_1:def 14;
  then
A9: [SP.i,SP.j] in Indices M by A1,A7,Th17;
  S*(i,j)=M*(SP.i,SP.j) by A7,Def1;
  hence thesis by A9,A8,MATRIX_1:def 6;
end;
