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 Th25:
  Indices block_diagonal(F1,d1) is Subset of Indices block_diagonal(F1^F2,d2)
proof
  set B1=block_diagonal(F1,d1);
  set B2=block_diagonal(F1^F2,d2);
  Indices B1 c= Indices B2
  proof
    (Len F1)^(Len F2)=Len (F1^F2) by Th14;
    then Sum Len F1+Sum Len F2=Sum Len (F1^F2) by RVSUM_1:75;
    then 0+Sum Len F1 <= Sum Len (F1^F2) by XREAL_1:6;
    then
A1: Seg Sum Len F1 c= Seg Sum Len (F1^F2) by FINSEQ_1:5;
A2: dom B2=Seg len B2 by FINSEQ_1:def 3;
    (Width F1)^(Width F2)=Width (F1^F2) by Th18;
    then Sum Width F1+Sum Width F2= Sum Width (F1^F2) by RVSUM_1:75;
    then 0+Sum Width F1<= Sum Width(F1^F2) by XREAL_1:6;
    then
A3: Seg Sum Width F1 c= Seg Sum Width (F1^F2) by FINSEQ_1:5;
A4: len B1=Sum Len F1 by Def5;
    let x be object;
    assume x in Indices B1;
    then
A5: ex i,j be object st i in dom B1 & j in Seg width B1 & x=[i,j] by
ZFMISC_1:def 2;
A6: dom B1=Seg len B1 by FINSEQ_1:def 3;
A7: width B1=Sum Width F1 by Def5;
A8: width B2=Sum Width (F1^F2) by Def5;
    len B2=Sum Len (F1^F2) by Def5;
    hence thesis by A5,A6,A2,A4,A8,A7,A1,A3,ZFMISC_1:87;
  end;
  hence thesis;
end;
