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 Th25:
  i in Seg n & j in Seg n & nt.i = nt.j implies Line(Segm(A,nt,mt)
  ,i) = Line(Segm(A,nt,mt),j)
proof
  set S=Segm(A,nt,mt);
  set Li=Line(S,i);
  set Lj=Line(S,j);
  assume that
A1: i in Seg n and
A2: j in Seg n and
A3: nt.i=nt.j;
A4: now
    let k such that
A5: 1<=k and
A6: k<=width S;
A7: k in Seg width S by A5,A6;
    then [i,k] in [:Seg n,Seg width S:] by A1,ZFMISC_1:87;
    then [i,k] in Indices S by MATRIX_0:25;
    then
A8: S*(i,k)=A*(nt.i,mt.k) by Def1;
    [j,k] in [:Seg n,Seg width S:] by A2,A7,ZFMISC_1:87;
    then [j,k] in Indices S by MATRIX_0:25;
    then
A9: S*(j,k)=A*(nt.j,mt.k) by Def1;
    S*(i,k) =Li.k by A7,MATRIX_0:def 7;
    hence Li.k=Lj.k by A3,A7,A8,A9,MATRIX_0:def 7;
  end;
A10: len Lj=width S by MATRIX_0:def 7;
  len Li=width S by MATRIX_0:def 7;
  hence thesis by A10,A4;
end;
