
theorem Th22:
  for K being Field,M1,M2 being Matrix of K st width M1=len M2 &
  len M1>0 holds 0.(K,len M1,width M1)*M2=0.(K,len M1,width M2)
proof
  let K be Field,M1,M2 be Matrix of K;
  assume that
A1: width M1=len M2 and
A2: len M1>0;
A4: len (0.(K,len M1,width M1))=len M1 by MATRIX_0:def 2;
  then
A5: width (0.(K,len M1,width M1))=width M1 by A2,MATRIX_0:20;
  then
A6: len ((0.(K,len M1,width M1))*M2)=len (0.(K,len M1,width M1)) by A1,
MATRIX_3:def 4;
A7: width ((0.(K,len M1,width M1))*M2)=width M2 by A1,A5,MATRIX_3:def 4;
  set B=(0.(K,len M1,width M1))*M2;
A8: width -((0.(K,len M1,width M1))*M2)=width ((0.(K,len M1,width M1))*M2)
  by MATRIX_3:def 2;
  (0.(K,len M1,width M1))*M2 =((0.(K,len M1,width M1)+(0.(K,len M1,width
  M1))))*M2 by MATRIX_3:4
    .=((0.(K,len M1,width M1)))*M2+((0.(K,len M1,width M1)))*M2 by A1,A4
,A5,MATRIX_4:63;
  then
  len -((0.(K,len M1,width M1))*M2)=len ((0.(K,len M1,width M1))*M2) & 0.
  (K, len M1,width M2)=B+B+(-B) by A4,A6,A7,MATRIX_3:def 2,MATRIX_4:2;
  then 0.(K,len M1,width M2) =B+(B-B) by A8,MATRIX_3:3
    .=(0.(K,len M1,width M1))*M2 by A6,A8,MATRIX_4:20;
  hence thesis;
end;
