 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 LmSign1E:
  for n, i, j 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
  holds Delete(M,i,j) is Matrix of n, INT
  proof
    let n, i, j be Nat, M be Matrix of n+1,F_Real;
    assume that
    A1: 0 < n and
    A2:M is Matrix of n+1, INT and
    A3: [i, j] in Indices M;
    set M0 = Delete(M,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;
    for x being object st x in rng M0 holds
    ex p being FinSequence of INT st x = p & len p = n
    proof
      let x be object;
      assume S1: x in rng M0;
      then reconsider p = x as FinSequence of the carrier of F_Real
      by FINSEQ_2:def 3;
      S3: len p = n by S1,X39,MATRIX_0:def 2;
      for z being object st z in rng p holds z in INT
      proof
        let z be object;
        assume z in rng p;
        then consider j1 be object such that
        S4: j1 in dom p & z = p.j1 by FUNCT_1:def 3;
        S5: j1 in Seg n by S3,S4,FINSEQ_1:def 3;
        reconsider j1 as Nat by S4;
        consider i1 be object such that
        S6: i1 in dom M0 & x = M0.i1 by S1,FUNCT_1:def 3;
        reconsider i1 as Nat by S6;
        S8: [i1,j1] in Indices M0 by D2,S5,S6,ZFMISC_1:87;
        then consider q be FinSequence of F_Real such that
        S9: q = M0.i1 & M0*(i1,j1) = q.j1 by MATRIX_0:def 5;
        M0*(i1,j1) is Element of INT by A1,A2,A3,S8,LmSign1F;
        hence z in INT by S4,S6,S9;
      end; then
      rng p c= INT;
      then p is FinSequence of INT by FINSEQ_1:def 4;
      hence thesis by S3;
    end;
    hence thesis by A1,D2,MATRIX_0:9,MATRIX_0:20;
  end;
