reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,a1,a2 for Element of K,
  D for non empty set,
  d,d1,d2 for Element of D,
  M,M1,M2 for (Matrix of D),
  A,A1,A2,B1,B2 for (Matrix of K),
  f,g for FinSequence of NAT;
reserve F,F1,F2 for FinSequence_of_Matrix of D,
  G,G9,G1,G2 for FinSequence_of_Matrix of K;

theorem Th31:
  i in dom F implies F.i = Segm(block_diagonal(F,d),(Seg Sum((Len
F) |i))\Seg Sum((Len F) | (i-'1)), (Seg Sum((Width F) |i))\Seg Sum((Width F)
| (i-'1)
  ))
proof
  assume
A1: i in dom F;
  set L=Len F;
  set Fi=F.i;
A2: dom F=dom L by Def3;
  then
A3: len Fi=L.i by A1,Def3;
  set SL=Sum(L|i);
  set SL1=Sum(L| (i-'1));
A4: SL1+L.i=SL by A1,A2,Lm2;
  reconsider FI=Fi as Matrix of len Fi,width Fi,D by MATRIX_0:51;
  set B=block_diagonal(F,d);
  set W=Width F;
  set SW1=Sum(W| (i-'1));
  set SW=Sum(W|i);
A5: dom F=dom W by Def4;
  then
A6: SW1+W.i=SW by A1,Lm2;
  SL1 <= SL by NAT_D:35,POLYNOM3:18;
  then Seg SL1 c= Seg SL by FINSEQ_1:5;
  then
A7: card (Seg SL\Seg SL1) = card Seg SL-card Seg SL1 by CARD_2:44
    .= SL-card Seg SL1 by FINSEQ_1:57
    .= SL1+(L.i)-SL1 by A4,FINSEQ_1:57
    .= len Fi by A1,A2,Def3;
  SW1 <=SW by NAT_D:35,POLYNOM3:18;
  then Seg SW1 c= Seg SW by FINSEQ_1:5;
  then
A8: card (Seg SW\Seg SW1) = card Seg SW-card Seg SW1 by CARD_2:44
    .= SW-card Seg SW1 by FINSEQ_1:57
    .= SW1+(W.i)-SW1 by A6,FINSEQ_1:57
    .= width Fi by A1,A5,Def4;
A9: width Fi=W.i by A1,A5,Def4;
  now
A10: Indices FI = Indices Segm(B,Seg SL\Seg SL1,Seg SW\Seg SW1) by A7,A8,
MATRIX_0:26;
A11: dom FI=Seg (L.i) by A3,FINSEQ_1:def 3;
    let j,k such that
A12: [j,k] in Indices FI;
    k in Seg (W.i) by A9,A12,ZFMISC_1:87;
    then
A13: Sgm(Seg SW\Seg SW1).k=k+SW1 by A6,MATRIX15:8;
    j in dom FI by A12,ZFMISC_1:87;
    then Sgm(Seg SL\Seg SL1).j=j+SL1 by A4,A11,MATRIX15:8;
    hence
    Segm(B,Seg SL\Seg SL1,Seg SW\Seg SW1)*(j,k) = B*(j+SL1,k+SW1) by A12,A10
,A13,MATRIX13:def 1
      .= Fi*(j,k) by A1,A12,Th30;
  end;
  hence thesis by A7,A8,MATRIX_0:27;
end;
