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 Th96:
  the_rank_of M = 1 iff (ex i,j st [i,j] in Indices M & M*(i,j) <>
  0.K) & for i0,j0,n0,m0 st i0<>j0 & n0<>m0 & [:{i0,j0},{n0,m0}:] c= Indices M
  holds Det EqSegm(M,{i0,j0},{n0,m0}) = 0.K
proof
  consider P,Q such that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q and
A3: card P = the_rank_of M and
A4: Det EqSegm(M,P,Q)<>0.K by Def4;
  thus the_rank_of M = 1 implies (ex i,j st [i,j] in Indices M & M*(i,j)<>0.K)
  & for i0,j0,n0,m0 st i0<>j0 & n0<>m0 & [:{i0,j0},{n0,m0}:] c= Indices M holds
  Det EqSegm(M,{i0,j0},{n0,m0}) = 0.K
  proof
    assume
A5: the_rank_of M = 1;
    hence ex i,j st [i,j] in Indices M & M*(i,j) <> 0.K by Th94;
    let i0,j0,n0,m0 such that
A6: i0<>j0 and
A7: n0<>m0 and
A8: [:{i0,j0},{n0,m0}:] c= Indices M;
A9: card {n0,m0}=2 by A7,CARD_2:57;
    assume
A10: Det EqSegm(M,{i0,j0},{n0,m0}) <> 0.K;
    card {i0,j0} =2 by A6,CARD_2:57;
    hence thesis by A5,A8,A10,A9,Def4;
  end;
  assume ex i,j st [i,j] in Indices M & M*(i,j) <> 0.K;
  then
A11: the_rank_of M>0 by Th94;
  assume
A12: for i0,j0,n0,m0 st i0<>j0 & n0<>m0 & [:{i0,j0},{n0,m0}:]c=Indices M
  holds Det EqSegm(M,{i0,j0},{n0,m0}) = 0.K;
  assume the_rank_of M <> 1;
  then card P>1 by A11,A3,NAT_1:25;
  then card P>=1+1 by NAT_1:13;
  then consider P1 be finite Subset of P such that
A13: card P1=2 by FINSEQ_4:72;
  not 0 in P1;
  then reconsider P1 as without_zero finite Subset of NAT by MEASURE6:def 2
,XBOOLE_1:1;
  consider Q1 such that
A14: Q1 c=Q and
A15: card P1=card Q1 and
A16: Det EqSegm(M,P1,Q1)<>0.K by A2,A4,Th65;
  consider n,m be object such that
A17: n <> m and
A18: Q1 = {n,m} by A13,A15,CARD_2:60;
A19: n in Q1 by A18,TARSKI:def 2;
  m in Q1 by A18,TARSKI:def 2;
  then reconsider n,m as non zero Element of NAT by A19;
  consider i,j be object such that
A20: i <> j and
A21: P1 = {i,j} by A13,CARD_2:60;
A22: i in P1 by A21,TARSKI:def 2;
  j in P1 by A21,TARSKI:def 2;
  then reconsider i,j as non zero Element of NAT by A22;
  [:P1,Q1:] c= [:P,Q:] by A14,ZFMISC_1:96;
  then [:{i,j},{n,m}:] c= Indices M by A1,A21,A18;
  hence thesis by A12,A20,A21,A16,A17,A18;
end;
