theorem Th69:
  i in Seg width G & width G = m+1 & m>0 & n in dom G & 1<=k & k<i
  implies DelCol(G,i)*(n,k) = G*(n,k) & k 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: 1<=k & k<i;
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) by A1,A2,A4,Th67;
  hence DelCol(G,i)*(n,k) = Col(G,k).n by A3,A6,A5,Def8
    .= G*(n,k) by A3,Def8;
  thus thesis by A1,A2,A4,Th67;
end;
