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

theorem Th5:
  for A be Matrix of n,K st i in Seg n & j in Seg n holds Delete(a*
  A,i,j)=a*Delete(A,i,j)
proof
  let A 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*A,i,j) = Deleting(a*A,i,j) by A1,LAPLACE:def 1
    .= Segm(a*A,Seg n\{i},Seg n\{j}) by MATRIX13:58
    .= a*Segm(A,Seg n\{i},Seg n\{j}) by A2,A3,MATRIX13:63
    .= a*Deleting(A,i,j) by MATRIX13:58
    .= a*Delete(A,i,j) by A1,LAPLACE:def 1;
end;
