reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th18:
  for K being Ring
  for m being Nat,A,C being Matrix of K st width A>0
  holds A*(0.(K,width A,m)) = 0.(K,len A,m)
proof
  let K be Ring;
  let m be Nat, A,C be Matrix of K;
  assume
A1: width A>0;
A3: len (0.(K,width A,m))= width A by MATRIX_0:def 2;
  then m=width 0.(K,width A,m) by A1,MATRIX_0:20;
  then
A4: width (A*((0.(K,width A,m))))=m by A3,MATRIX_3:def 4;
  width (0.(K,width A,m)) = m by A1,A3,MATRIX_0:20;
  then
A5: (A * (0.(K,width A,m))) + (A * (0.(K,width A,m)))
   = A * (0.(K,width A,m) + 0.(K,width A,m)) by A3,MATRIX_4:62
  .= A * (0.(K,width A,m)) by MATRIX_3:4;
  len (A*(0.(K,width A,m)))=len A by A3,MATRIX_3:def 4;
  hence thesis by A4,A5,MATRIX_4:6;
end;
