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 Th33:
  M = Segm(block_diagonal(F^<*M*>,d),Seg (len M+Sum Len F)\Seg Sum
  Len F, Seg (width M+Sum Width F)\Seg Sum Width F)
proof
  set FM=F^<*M*>;
  set L=Len FM;
  set W=Width FM;
  set 1F=1+len F;
A1: len FM=1F by FINSEQ_2:16;
  len L=len FM by CARD_1:def 7;
  then
A2: L|1F=L by A1,FINSEQ_1:58;
  len W=len FM by CARD_1:def 7;
  then
A3: W|1F=W by A1,FINSEQ_1:58;
  1F in Seg (len F+1) by FINSEQ_1:4;
  then
A4: 1F in dom FM by A1,FINSEQ_1:def 3;
A5: W=Width F^Width<*M*> by Th18;
  len Width F=len F by CARD_1:def 7;
  then
A6: W| (len F)=Width F by A5,FINSEQ_5:23;
  Width <*M*>=<*width M*> by Th19;
  then
A7: Sum W=Sum Width F+width M by A5,RVSUM_1:74;
A8: L = Len F ^ Len <*M*> by Th14;
  len Len F=len F by CARD_1:def 7;
  then
A9: L| (len F)=Len F by A8,FINSEQ_5:23;
  Len <*M*>=<*len M*> by Th15;
  then
A10: Sum L=Sum Len F+len M by A8,RVSUM_1:74;
A11: FM.1F=M by FINSEQ_1:42;
  1F-'1=1F-1 by NAT_D:37;
  hence thesis by A9,A6,A10,A7,A4,A11,A3,A2,Th31;
end;
