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;
