 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 LmSign1B:
  for D, E being non empty set, n, m, i, j being Nat, M being Matrix of n,m,D
  st 0 < n & M is Matrix of n,m,E & [i, j] in Indices M
  holds M*(i,j) is Element of E
  proof
    let D, E be non empty set, n, m, i, j be Nat,
    M be Matrix of n,m,D;
    assume that
    A1: 0 < n and
    A2: M is Matrix of n, m, E and
    A3: [i, j] in Indices M;
    consider m1 be Nat such that
    A4: for x being object st x in rng M
    ex q being FinSequence of E st x = q & len q = m1 by MATRIX_0:9,A2;
    consider p be FinSequence of D such that
    A5: p = M.i & M*(i,j) = p.j by A3,MATRIX_0:def 5;
    A6: i in dom M & j in Seg width M by A3,ZFMISC_1:87;
    then
    A7: p in rng M by FUNCT_1:3,A5;
    ex q being FinSequence of E st p = q & len q = m1 by A4,A5,A6,FUNCT_1:3;
    then
    A50: rng p c= E by FINSEQ_1:def 4;
    len p = m by A7,MATRIX_0:def 2;
    then len p = width M by A1,MATRIX_0:23;
    then j in dom p by FINSEQ_1:def 3,A6;
    hence thesis by A5,A50,FUNCT_1:3;
  end;
