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 Th70:
  i in Seg width G & width G = m+1 & m>0 & n in dom G & i<=k & k<=
  m implies DelCol(G,i)*(n,k) = G*(n,k+1) & k+1 in Seg width G
proof
  assume that
A1: i in Seg width G and
A2: width G = m+1 & m>0 and
A3: n in dom G and
A4: i<=k & k<=m;
A5: len (DelCol(G,i)) = len G by Def13;
A6: dom G = Seg len G & Seg len(DelCol(G,i)) = dom(DelCol(G,i)) by
FINSEQ_1:def 3;
  Col(DelCol(G,i),k)=Col(G,k+1) by A1,A2,A4,Th68;
  hence DelCol(G,i)*(n,k) = Col(G,k+1).n by A3,A6,A5,Def8
    .= G*(n,k+1) by A3,Def8;
  thus thesis by A1,A2,A4,Th68;
end;
