reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;

theorem Th94:
  the_rank_of M > 0 iff ex i,j st [i,j] in Indices M & M*(i,j) <> 0.K
proof
  set r=the_rank_of M;
  thus r > 0 implies ex i,j st [i,j] in Indices M & M*(i,j) <> 0.K
  proof
    consider P,Q such that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q and
A3: card P = r and
A4: Det EqSegm(M,P,Q)<>0.K by Def4;
    assume r>0;
    then consider x being object such that
A5: x in P by A3,CARD_1:27,XBOOLE_0:def 1;
    reconsider x as non zero Element of NAT by A5;
    {x} c= P by A5,ZFMISC_1:31;
    then consider Q1 such that
A6: Q1 c= Q and
A7: card {x} = card Q1 and
A8: Det EqSegm(M,{x},Q1) <> 0.K by A2,A4,Th65;
    consider y being object such that
A9: {y}=Q1 by A7,CARD_2:42;
    y in {y} by TARSKI:def 1;
    then reconsider y as non zero Element of NAT by A9;
    take x,y;
    y in Q1 by A9,TARSKI:def 1;
    then [x,y] in [:P,Q:] by A5,A6,ZFMISC_1:87;
    hence [x,y] in Indices M by A1;
A10: card { x } = 1 by CARD_1:30;
    EqSegm(M,{x},Q1) = Segm(M,{x},{y}) by A7,A9,Def3
      .= <*<* M*(x,y) *>*> by Th44;
    hence thesis by A8,A10,MATRIX_3:34;
  end;
  given i,j such that
A11: [i,j] in Indices M and
A12: M*(i,j) <> 0.K;
A13: j in Seg width M by A11,ZFMISC_1:87;
  Indices M = [:Seg len M,Seg width M:] by FINSEQ_1:def 3;
  then
A14: i in Seg len M by A11,ZFMISC_1:87;
  then reconsider i,j as non zero Element of NAT by A13;
A15: card { i } = 1 by CARD_1:30;
A16: card {j}=1 by CARD_1:30;
  then EqSegm(M,{i},{j}) = Segm(M,{i},{j}) by Def3,CARD_1:30
    .= <*<* M*(i,j) *>*> by Th44;
  then
A17: Det EqSegm(M,{i},{j})<>0.K by A12,A15,MATRIX_3:34;
A18: {j} c= Seg width M by A13,ZFMISC_1:31;
  {i} c= Seg len M by A14,ZFMISC_1:31;
  then [:{i},{j}:] c= Indices M by A15,A16,A18,Th67;
  hence thesis by A15,A16,A17,Def4;
end;
