reserve x,X for set,
        r,r1,r2,s for Real,
        i,j,k,m,n for Nat;

theorem Th2:
  for K be Field for M be diagonal Matrix of n,K holds M@ = M
proof
  let K be Field;
  let M be diagonal Matrix of n,K;
  for i,j st[i,j] in Indices M holds M*(i,j)=M@*(i,j)
  proof
    let i,j;
    assume A1: [i,j] in Indices M;
    then A2: [j,i] in Indices M by MATRIX_0:28;
    then A3: M@*(i,j)=M*(j,i) by MATRIX_0:def 6;
    per cases;
    suppose i=j;
      hence thesis by A1,MATRIX_0:def 6;
    end;
    suppose A4: i<>j;
      then M*(i,j)=0.K by A1,MATRIX_1:def 6;
      hence thesis by A2,A3,A4,MATRIX_1:def 6;
    end;
  end;
  hence thesis by MATRIX_0:27;
end;
