
theorem Th24:
  for K being Field,M1 being Matrix of K st len M1>0 holds 0.K*M1=
  0.(K,len M1,width M1)
proof
  let K be Field,M1 be Matrix of K;
A1: len (0.(K,len M1,width M1))=len M1 by MATRIX_0:def 2;
  assume len M1>0;
  then
A2: width (0.(K,len M1,width M1))=width M1 by A1,MATRIX_0:20;
A3: for i,j be Nat st [i,j] in Indices (0.(K,len M1,width M1)) holds ((0.K)*
  M1)*(i,j)=(0.(K,len M1,width M1))*(i,j)
  proof
    let i,j be Nat;
    assume
A4: [i,j] in Indices (0.(K,len M1,width M1));
    Indices (0.(K,len M1,width M1))= Indices M1 by A1,A2,MATRIX_4:55;
    then
A5: ((0.K)*M1)*(i,j)=(0.K)*(M1*(i,j)) by A4,MATRIX_3:def 5;
    (0.(K,len M1,width M1))*(i,j)=0.K by A4,MATRIX_3:1;
    hence thesis by A5;
  end;
  len (0.K*M1)=len M1 & width (0.K*M1)=width M1 by MATRIX_3:def 5;
  hence thesis by A1,A2,A3,MATRIX_0:21;
end;
