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
  width G = m+1 & m>0 & k in Seg m & n in dom G implies k in Seg
  width G & DelCol(G,width G)*(n,k) = G*(n,k) & width G in Seg width G
proof
  assume that
A1: width G = m+1 and
A2: m>0 and
A3: k in Seg m and
A4: n in dom G;
  k<=m by A3,FINSEQ_1:1;
  then
A5: k<width G by A1,NAT_1:13;
  1<=width G by A1,A2,SEQM_3:43;
  then
A6: width G in Seg width G by FINSEQ_1:1;
  1<=k by A3,FINSEQ_1:1;
  hence thesis by A1,A2,A4,A6,A5,Th69;
end;
