reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D;
reserve M,M1,M2 for Matrix of D;
reserve f for FinSequence of D;
reserve i,j,i1,j1 for Nat;
reserve k for Nat, G for Matrix of D;
reserve x,y,x1,x2,y1,y2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  s,s2 for FinSequence,
  a,b,c,d for Element of D,
  q,r for FinSequence of D,
  a9,b9 for Element of D;
reserve m for Nat;

theorem Th67:
  i in Seg width G & width G = m+1 & m>0 & 1<=k & k<i implies Col(
  DelCol(G,i),k) = Col(G,k) & k in Seg width DelCol(G,i) & k in Seg width G
proof
  set N = DelCol(G,i);
  assume that
A1: i in Seg width G and
A2: width G = m+1 and
A3: m>0 and
A4: 1<=k and
A5: k<i;
A6: width N = m by A1,A2,Th63;
A7: 1<width G by A2,A3,SEQM_3:43;
A8: len N = len G by Def13;
  i<=m+1 by A1,A2,FINSEQ_1:1;
  then
A9: k<m+1 by A5,XXREAL_0:2;
  then
A10: k in Seg width G by A2,A4,FINSEQ_1:1;
A11: len Col(G,k)=len G by Def8;
A12: len Col(N,k) = len N by Def8;
A13: k<=m by A9,NAT_1:13;
  then
A14: k in Seg width N by A4,A6,FINSEQ_1:1;
  now
    let j be Nat;
A15: dom N = Seg len N & dom G = Seg len G by FINSEQ_1:def 3;
    assume 1<=j & j<=len Col(N,k);
    then
A16: j in dom N by A12,FINSEQ_3:25;
    hence Col(N,k).j= N*(j,k) by Def8
      .= Del(Line(G,j),i).k by A1,A14,A7,A8,A16,A15,Th66
      .= Line(G,j).k by A5,FINSEQ_3:110
      .= Col(G,k).j by A10,A8,A16,A15,Th42;
  end;
  hence thesis by A2,A4,A6,A9,A13,A12,A11,A8,FINSEQ_1:1,14;
end;
