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;
reserve S,S1,S2 for FinSequence_of_Square-Matrix of D,
  R,R1,R2 for FinSequence_of_Square-Matrix of K;

theorem
  for A be Matrix of n,K st i in dom A & j in Seg n holds Deleting(
block_diagonal(R^<*A*>,a),i+Sum Len R,j+Sum Len R)= block_diagonal(R^<*Deleting
  (A,i,j)*>,a)
proof
  let A be Matrix of n,K such that
A1: i in dom A and
A2: j in Seg n;
  n <> 0 by A2;
  then
A3: n>=1 by NAT_1:14;
  set jS=j+Sum Len R;
  set iS=i+Sum Len R;
  set AA=<*A*>;
  set b=block_diagonal(R,a);
  set B=<*b*>;
  set LRA=Sum Len(R^AA);
  set LBA=Sum Len(B^AA);
A4: width A=n by MATRIX_0:24;
A5: Len R=Width R by Th46;
A6: len A=n by MATRIX_0:24;
  then
A7: dom A=Seg n by FINSEQ_1:def 3;
A8: Width AA=<*width A*> by Th19;
A9: len b=Sum Len R by Def5;
A10: Len(B^AA)=Width(B^AA) by Th46;
A11: Len AA=<*len A*> by Th15;
  then
A12: Sum Len AA=len A by RVSUM_1:73;
  Len(B^AA)=(Len B)^Len AA by Th14;
  then
A13: LBA=len A+Sum Len B by A11,RVSUM_1:74;
A14: Width B=<*width b*> by Th19;
  then
A15: Sum Width B=width b by RVSUM_1:73;
  Width(B^AA)=(Width B)^Width AA by Th18;
  then
A16: LBA=Sum Width AA+width b by A14,A10,RVSUM_1:76;
  Len B=<*len b*> by Th15;
  then
A17: Sum Len B=len b by RVSUM_1:73;
  per cases by A3,XXREAL_0:1;
  suppose
A18: n=1;
    then
A19: i=1 by A1,A7,FINSEQ_1:2,TARSKI:def 1;
A20: j=1 by A2,A18,FINSEQ_1:2,TARSKI:def 1;
    len Deleting(A,i,j) = 1-'1 by A1,A18,LAPLACE:2
      .= 0 by XREAL_1:232;
    then
A21: Deleting(A,i,j)={};
    thus Deleting(block_diagonal(R^AA,a),iS,jS) = Deleting(block_diagonal(B^AA
    ,a),iS,jS) by Th35
      .= Segm(block_diagonal(B^AA,a),Seg LBA\{iS},Seg LBA\{jS}) by MATRIX13:58
      .= Segm(block_diagonal(B^AA,a),Seg Sum Len B,Seg LBA\{jS}) by A6,A9,A13
,A17,A18,A19,FINSEQ_1:10
      .= Segm(block_diagonal(B^AA,a),Seg Sum Len B,Seg Sum Width B) by A6,A4,A9
,A11,A8,A13,A16,A17,A12,A15,A18,A20,FINSEQ_1:10
      .= b by A17,A15,Th32
      .= block_diagonal(B,a) by Th34
      .= block_diagonal(B^<*Deleting(A,i,j)*>,a) by A21,Th40
      .= block_diagonal(R^<*Deleting(A,i,j)*>,a) by Th35;
  end;
  suppose
    n>1;
    then
A22: width A=width DelLine(A,i) by A6,LAPLACE:4;
    thus Deleting(block_diagonal(R^AA,a),iS,jS) = DelCol(DelLine(
    block_diagonal(R^AA,a),iS),jS)
      .= DelCol(block_diagonal(R^<*DelLine(A,i)*>,a),jS) by A1,A22,Th42
      .= block_diagonal(R^<*DelCol(DelLine(A,i),j)*>,a)by A2,A4,A5,A22,Th44
      .= block_diagonal(R^<*Deleting(A,i,j)*>,a);
  end;
end;
