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

theorem Th14:
  for K being Ring
  for A being Matrix of K st n>0 holds 
  0.(K,n,len A)*A = 0.(K,n,width A)
proof
  let K be Ring;
  let A be Matrix of K;
  assume
A1: n>0;
A3: len 0.(K,n,len A)=n by MATRIX_0:def 2;
  then
A4: width 0.(K,n,len A)=len A by A1,MATRIX_0:20;
  then
A5: len ((0.(K,n,len A))*A)=n by A3,MATRIX_3:def 4;
A6: width ((0.(K,n,len A))*A)=width A by A4,MATRIX_3:def 4;
  0.(K,n,len A)*A + 0.(K,n,len A)*A =(0.(K,n,len A)+ 0.(K,n,len A))*A by
A3,A4,MATRIX_4:63
    .=0.(K,n,len A)*A by MATRIX_3:4;
  hence thesis by A5,A6,MATRIX_4:6;
end;
