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 Th27:
  [i,j] in Indices block_diagonal(F2,d1) iff i>0 & j>0 & [i+Sum
  Len F1,j+Sum Width F1] in Indices block_diagonal(F1^F2,d2)
proof
  set B2=block_diagonal(F2,d1);
  set B12=block_diagonal(F1^F2,d2);
A1: dom B12=Seg len B12 by FINSEQ_1:def 3;
A2: len B12=Sum Len (F1^F2) by Def5;
  (Len F1)^Len F2=Len (F1^F2) by Th14;
  then
A3: Sum Len F1+Sum Len F2=Sum Len (F1^F2) by RVSUM_1:75;
A4: len B2=Sum Len F2 by Def5;
  (Width F1)^Width F2=Width (F1^F2) by Th18;
  then
A5: Sum Width F1+Sum Width F2= Sum Width (F1^F2) by RVSUM_1:75;
A6: width B12=Sum Width (F1^F2) by Def5;
A7: width B2=Sum Width F2 by Def5;
A8: dom B2=Seg len B2 by FINSEQ_1:def 3;
  hereby
    assume
A9: [i,j] in Indices B2;
    then
A10: j in Seg width B2 by ZFMISC_1:87;
    then
A11: j+Sum Width F1 in Seg width B12 by A6,A7,A5,FINSEQ_1:60;
A12: i in Seg len B2 by A8,A9,ZFMISC_1:87;
    then i +Sum Len F1 in Seg len B12 by A2,A4,A3,FINSEQ_1:60;
    hence i>0 & j>0 & [i+Sum Len F1,j+Sum Width F1] in Indices B12 by A1,A12
,A10,A11,ZFMISC_1:87;
  end;
  assume that
A13: i>0 and
A14: j>0 and
A15: [i+Sum Len F1,j+Sum Width F1] in Indices B12;
  i +Sum Len F1 in Seg len B12 by A1,A15,ZFMISC_1:87;
  then
A16: i in Seg len B2 by A2,A4,A3,A13,FINSEQ_1:61;
  j +Sum Width F1 in Seg width B12 by A15,ZFMISC_1:87;
  then
A17: j in Seg width B2 by A6,A7,A5,A14,FINSEQ_1:61;
  dom B2=Seg len B2 by FINSEQ_1:def 3;
  hence thesis by A16,A17,ZFMISC_1:87;
end;
