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 Th30:
  for i,j,k st i in dom F & [j,k] in Indices F.i holds [j+Sum ((
Len F) | (i-'1)), k+Sum((Width F) | (i-'1))] in Indices block_diagonal(F,d) &
(F.i)*(j,k) = block_diagonal(F,d)*(j+Sum ((Len F) | (i-'1)), k+Sum((Width F) |
(i-'1)))
proof
  let i,j,k such that
A1: i in dom F and
A2: [j,k] in Indices F.i;
  set Fi=F.i;
A3: k in Seg width Fi by A2,ZFMISC_1:87;
  set L=Len F;
  len F=len L by CARD_1:def 7;
  then
A4: L| (len F)=L by FINSEQ_1:58;
A5: i<=len F by A1,FINSEQ_3:25;
  then Sum (L|i)<=Sum(L| (len F)) by POLYNOM3:18;
  then
A6: Seg Sum (L|i) c= Seg Sum L by A4,FINSEQ_1:5;
A7: dom F=dom L by Def3;
  then
A8: L.i=L/.i by A1,PARTFUN1:def 6;
  set W=Width F;
A9: dom F=dom W by Def4;
  then
A10: W.i=W/.i by A1,PARTFUN1:def 6;
  set kS=k+Sum(W| (i-'1));
  W.i=width Fi by A1,A9,Def4;
  then
A11: kS in Seg Sum (W|i) by A1,A9,A3,A10,Th10;
  then
A12: kS <= Sum(W|i) by FINSEQ_1:1;
  len F=len W by CARD_1:def 7;
  then
A13: W| (len F)=W by FINSEQ_1:58;
  set B=block_diagonal(F,d);
A14: len B=Sum L by Def5;
A15: width B=Sum W by Def5;
  set jS=j+Sum(L| (i-'1));
  j in dom Fi by A2,ZFMISC_1:87;
  then
A16: j in Seg len Fi by FINSEQ_1:def 3;
  Sum(W|i)<=Sum (W| (len F)) by A5,POLYNOM3:18;
  then
A17: Seg Sum (W|i) c= Seg Sum W by A13,FINSEQ_1:5;
  Sum(L| (i-'1))+0<=jS by XREAL_1:6;
  then
A18: jS-'Sum(L| (i-'1))=jS-Sum(L| (i-'1)) by XREAL_1:233;
A19: L.i=len Fi by A1,A7,Def3;
  then
A20: min(L,jS)=i by A1,A7,A16,A8,Th10;
  jS in Seg Sum (L|i) by A1,A7,A16,A8,A19,Th10;
  then [jS,kS] in [:Seg len B,Seg width B:] by A14,A15,A11,A6,A17,ZFMISC_1:87;
  hence
A21: [jS,kS] in Indices B by FINSEQ_1:def 3;
  0<k by A3;
  then
A22: Sum(W| (i-'1))+0<kS by XREAL_1:6;
  then kS-'Sum(W| (i-'1))=kS- Sum(W| (i-'1)) by XREAL_1:233;
  hence thesis by A20,A21,A22,A12,A18,Def5;
end;
