
theorem LmSign1E:
  for n, i, j being Nat, M being Matrix of n+1, F_Real
  for L being Matrix of n+1, F_Rat
  st 0 < n & M = L  & [i, j] in Indices M
  holds Delete(M,i,j) = Delete(L,i,j)
  proof
    let n, i, j be Nat, M be Matrix of n+1,F_Real;
    let L be Matrix of n+1,F_Rat;
    assume that
    A1: 0 < n and
    A2: M = L and
    A3: [i, j] in Indices M;
    set M0 = Delete(M,i,j);
    set L0 = Delete(L,i,j);
    X39: (n+1)-'1 = n by NAT_D:34; then
    D2: len M0 = n & width M0 = n & Indices M0 = [:(Seg n),(Seg n):]
    by MATRIX_0:24;
    BD2: len L0 = n & width L0 = n & Indices L0 = [:(Seg n),(Seg n):]
    by MATRIX_0:24,X39;
    for i1,j1 being Nat st [i1,j1] in Indices M0
    holds M0*(i1,j1) = L0*(i1,j1) by LmSign1F,A1,A2,A3;
    hence thesis by BD2,D2,ZMATRLIN:4;
  end;
