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;

theorem Th30:
  i in Seg m & j in Seg m & mt.i = mt.j & i<>j implies Det Segm(M,
  mt1,mt) = 0.K
proof
  assume that
A1: i in Seg m and
A2: j in Seg m and
A3: mt.i = mt.j and
A4: i<>j;
A5: i<j or j<i by A4,XXREAL_0:1;
  set S = Segm(M,mt1,mt);
A6: width S=m by MATRIX_0:24;
  then
A7: Col(S,j)=Line(S@,j) by A2,MATRIX_0:59;
  Col(S,i)=Line(S@,i) by A1,A6,MATRIX_0:59;
  hence 0.K = Det (S@) by A1,A2,A3,A7,A5,Th29,MATRIX11:50
    .= Det S by MATRIXR2:43;
end;
