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
  the_rank_of M = the_rank_of (M@)
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;
A5: [:Q,P:] c= Indices (M@) by A1,A2,Th69;
  consider P1,Q1 such that
A6: [:P1,Q1:] c= Indices (M@) and
A7: card P1 = card Q1 and
A8: card P1 = the_rank_of (M@) and
A9: Det EqSegm(M@,P1,Q1)<>0.K by Def4;
A10: [:Q1,P1:] c= Indices M by A6,A7,Th69;
  then Det EqSegm(M,Q1,P1)<>0.K by A7,A9,Th70;
  then
A11: the_rank_of M >= the_rank_of (M@) by A7,A8,A10,Def4;
  Det EqSegm(M@,Q,P)<>0.K by A1,A2,A4,Th70;
  then the_rank_of (M@) >= the_rank_of M by A2,A3,A5,Def4;
  hence thesis by A11,XXREAL_0:1;
end;
