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 Th52:
  Segm(M,Seg len M,Seg width M\{i}) = DelCol(M,i)
proof
  set SW=Seg width M;
  set Si=SW\{i};
  set SL=Seg len M;
  set SEGM=Segm(M,SL,Si);
  set D=DelCol(M,i);
  card SL=len M by FINSEQ_1:57;
  then
A1: len SEGM=len M by MATRIX_0:def 2;
A2: now
    let j such that
A3: 1<=j and
A4: j<=len M;
A5: j in Seg len M by A3,A4;
    then
A6: j in dom M by FINSEQ_1:def 3;
    Sgm SL=idseq len M by FINSEQ_3:48;
    then
A7: Sgm SL.j=j by A5,FINSEQ_2:49;
    len Line(M,j)=width M by MATRIX_0:def 7;
    then
A8: dom Line(M,j)=SW by FINSEQ_1:def 3;
A9: card SL=len M by FINSEQ_1:57;
    then
A10: Line(SEGM,j)=SEGM.j by A5,MATRIX_0:52;
    Line(SEGM,j) = Line(M,Sgm SL.j) * Sgm Si by A9,A5,Th47,XBOOLE_1:36;
    then SEGM.j = Del(Line(M,j),i) by A7,A10,A8,FINSEQ_3:def 2;
    hence SEGM.j=D.j by A6,MATRIX_0:def 13;
  end;
  len D=len M by MATRIX_0:def 13;
  hence thesis by A1,A2;
end;
