reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a for Element of K;

theorem Th4:
  for A,B be Matrix of n,K st i in Seg n & j in Seg n holds
  Delete(A+B,i,j) = Delete(A,i,j) + Delete(B,i,j)
proof
  let A,B be Matrix of n,K such that
A1: i in Seg n & j in Seg n;
  Seg n\{i} c= Seg n & Seg n\{j} c= Seg n by XBOOLE_1:36;
  then
A2: [:Seg n\{i},Seg n\{j}:] c= [:Seg n,Seg n:] by ZFMISC_1:96;
A3: Indices A = [:Seg n,Seg n:] by MATRIX_0:24;
  thus Delete(A+B,i,j) = Deleting(A+B,i,j) by A1,LAPLACE:def 1
    .= Segm(A+B,Seg n\{i},Seg n\{j}) by MATRIX13:58
    .= Segm(A,Seg n\{i},Seg n\{j})+Segm(B,Seg n\{i},Seg n\{j}) by A2,A3,Th3
    .= Deleting(A,i,j)+Segm(B,Seg n\{i},Seg n\{j}) by MATRIX13:58
    .= Deleting(A,i,j) + Deleting(B,i,j) by MATRIX13:58
    .= Delete(A,i,j) + Deleting(B,i,j) by A1,LAPLACE:def 1
    .= Delete(A,i,j) + Delete(B,i,j) by A1,LAPLACE:def 1;
end;
