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 Th29:
  i in Seg m & j in Seg m & mt.i = mt.j implies Col(Segm(A,nt,mt),
  i) = Col(Segm(A,nt,mt),j)
proof
  set S=Segm(A,nt,mt);
  set Ci=Col(S,i);
  set Cj=Col(S,j);
  assume that
A1: i in Seg m and
A2: j in Seg m and
A3: mt.i=mt.j;
A4: now
    let k such that
A5: 1<=k and
A6: k<=len S;
A7: k in Seg len S by A5,A6;
    then
A8: k in Seg n by MATRIX_0:def 2;
    then n <> 0;
    then
A9: width S = m by Th1;
    [k,j] in [:Seg n,Seg m:] by A2,A8,ZFMISC_1:87;
    then [k,j] in Indices S by A9,MATRIX_0:25;
    then
A10: S*(k,j)=A*(nt.k,mt.j) by Def1;
    [k,i] in [:Seg n,Seg m:] by A1,A8,ZFMISC_1:87;
    then [k,i] in Indices S by A9,MATRIX_0:25;
    then
A11: S*(k,i)=A*(nt.k,mt.i) by Def1;
A12: k in dom S by A7,FINSEQ_1:def 3;
    then S*(k,i) =Ci.k by MATRIX_0:def 8;
    hence Ci.k=Cj.k by A3,A12,A11,A10,MATRIX_0:def 8;
  end;
A13: len Cj=len S by MATRIX_0:def 8;
  len Ci=len S by MATRIX_0:def 8;
  hence thesis by A13,A4;
end;
