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;
