
theorem Th9:
  for K being Field, M being Matrix of K holds (1_K)*M=M
proof
  let K be Field, M be Matrix of K;
A1: for i,j being Nat st [i,j] in Indices M holds ((1_K)*M)*(i,j)=M*(i,j)
  proof
    let i,j be Nat;
A2: (1_K)*(M*(i,j))=M*(i,j);
    assume [i,j] in Indices M;
    hence thesis by A2,MATRIX_3:def 5;
  end;
  len ((1_K)*M)=len M & width ((1_K)*M)=width M by MATRIX_3:def 5;
  hence thesis by A1,MATRIX_0:21;
end;
