reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem Th1:
  for K being Ring, A being Matrix of K holds
    (0.(K,len A,len A))*A = 0.(K,len A,width A)
proof
  let K be Ring, A be Matrix of K;
  per cases by NAT_1:3;
  suppose
A1: len A>0;
    set B = (0.(K,len A,len A))*A;
A2: width -((0.(K,len A,len A))*A)=width ((0.(K,len A,len A))*A) by
MATRIX_3:def 2;
A3: len (0.(K,len A,len A))=len A by MATRIX_0:def 2;
    then
A4: width (0.(K,len A,len A))=len A by A1,MATRIX_0:20;
    then
A5: len ((0.(K,len A,len A))*A)=len A by A3,MATRIX_3:def 4;
A6: width ((0.(K,len A,len A))*A)=width A by A4,MATRIX_3:def 4;
    (0.(K,len A,len A))*A =(0.(K,len A,len A)+0.(K,len A,len A))*A by
MATRIX_3:4
      .=(0.(K,len A,len A))*A+(0.(K,len A,len A))*A by A3,A4,MATRIX_4:63;
    then
    len -((0.(K,len A,len A))*A)=len ((0.(K,len A,len A))*A) & 0.(K,len A,
    width A)=B+B+(-B) by A5,A6,MATRIX_3:def 2,MATRIX_4:2;
    then 0.(K,len A,width A) =B+(B-B) by A2,MATRIX_3:3
      .=(0.(K,len A,len A))*A by A5,A2,MATRIX_4:20;
    hence thesis;
  end;
  suppose
A7: len A = 0;
    then len (0.(K,len A,len A)) = 0 by MATRIX_0:def 2;
    then width (0.(K,len A,len A)) = len A by A7,MATRIX_0:def 3;
    then
A8: len ((0.(K,len A,len A))*A) = len ((0.(K,len A,len A))) by MATRIX_3:def 4;
    len ((0.(K,len A,len A))) = len A & len (0.(K,len A,width A)) = len A
    by MATRIX_0:def 2;
    hence thesis by A7,A8,CARD_2:64;
  end;
end;
