 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve s for FinSequence,
   V1,V2,V3 for finite-rank free Z_Module,
   f,f1,f2 for Function of V1,V2,
   g for Function of V2,V3,
   b1 for OrdBasis of V1,
   b2 for OrdBasis of V2,
   b3 for OrdBasis of V3,
   v1,v2 for Vector of V2,
   v,w for Element of V1;
 reserve p2,F for FinSequence of V1,
   p1,d for FinSequence of INT.Ring,
   KL for Linear_Combination of V1;

theorem LmSign1F:
  for n, i, j, k, m being Nat, M being Matrix of n+1, F_Real
  st 0 < n & M is Matrix of n+1, INT &
  [i, j] in Indices M & [k, m] in Indices (Delete(M,i,j))
  holds (Delete(M,i,j))*(k,m) is Element of INT
  proof
    let n, i, j, k, m be Nat, M be Matrix of n+1,F_Real;
    assume that
    0 < n and
    A2: M is Matrix of n+1, INT and
    A3: [i, j] in Indices M and
    A4: [k, m] in Indices (Delete(M,i,j));
    [i, j] in [:Seg (n+1),Seg (n+1):] by A3,MATRIX_0:24;
    then
    A5: i in Seg (n+1) & j in Seg (n+1) by ZFMISC_1:87;
    set M0 = Delete(M,i,j);
    (n+1)-'1 = n by NAT_D:34;
    then len M0 = n & width M0 = n & Indices M0 = [:(Seg n),(Seg n):]
    by MATRIX_0:24; then
    D5: k in Seg n & m in Seg n by A4,ZFMISC_1:87;
    then
    D3: k in Seg ((n+1)-'1 ) & m in Seg ((n+1)-'1 ) by NAT_D:34;
    FC0: 1<=k & k <= n & 1<=m & m <= n by FINSEQ_1:1,D5;
    then 1<=k & k+0 <= n+1 & 1<=m & m+0 <= n+1 by XREAL_1:7;
    then
    FC1: k in Seg (n+1) & m in Seg (n+1) by FINSEQ_1:1;
    1+0<=k+1 & k+1 <= n+1 & 1+0<=m+1 & m+1 <= n+1 by FC0,XREAL_1:6; then
    FC3: k+1 in Seg (n+1) & m+1 in Seg (n+1) by FINSEQ_1:1;
    per cases;
    suppose k < i & m < j; then
      F11: Delete(M,i,j)*(k,m) = M*(k,m) by LAPLACE:13,A5,D3;
      [k, m] in [:Seg (n+1),Seg (n+1):] by FC1,ZFMISC_1:87;
      then [k, m] in Indices M by MATRIX_0:24;
      hence thesis by A2,F11,LmSign1B;
    end;
    suppose k < i & m >= j; then
      F21: Delete(M,i,j)*(k,m) = M*(k,m+1) by LAPLACE:13,A5,D3;
      [k, m+1] in [:Seg (n+1),Seg (n+1):] by FC1,FC3,ZFMISC_1:87;
      then [k, m+1] in Indices M by MATRIX_0:24;
      hence thesis by A2,F21,LmSign1B;
    end;
    suppose k >= i & m < j; then
      F31: Delete(M,i,j)*(k,m) = M*(k+1,m) by LAPLACE:13,A5,D3;
      [k+1, m] in [:Seg (n+1),Seg (n+1):] by FC1,FC3,ZFMISC_1:87;
      then [k+1, m] in Indices M by MATRIX_0:24;
      hence thesis by A2,F31,LmSign1B;
    end;
    suppose k >= i & m >= j; then
      F41: Delete(M,i,j)*(k,m) = M*(k+1,m+1) by LAPLACE:13,A5,D3;
      [k+1, m+1] in [:Seg (n+1),Seg (n+1):] by FC3,ZFMISC_1:87;
      then [k+1, m+1] in Indices M by MATRIX_0:24;
      hence thesis by A2,F41,LmSign1B;
    end;
  end;
