
theorem
  for n, i, j being Nat, M being Matrix of n+1, F_Real
  st M is Matrix of n+1, F_Rat & [i, j] in Indices M
  holds Cofactor(M,i,j) in F_Rat
  proof
    let n, i, j be Nat, M be Matrix of n+1, F_Real such that
    AS: M is Matrix of n+1, F_Rat & [i, j] in Indices M;
    (n+1)-'1 = n by NAT_D:34;
    then reconsider DD= Delete(M,i,j) as Matrix of n,F_Real;
    Det DD in F_Rat
    proof
      per cases;
      suppose 0 < n;
        then DD is Matrix of n, F_Rat by LmSign1EX,AS;
        hence Det DD in F_Rat by LmGM11;
      end;
      suppose not 0 < n;
        then n = 0;
        then Det DD = 1.F_Real by MATRIXR2:41
        .= 1;
        hence Det DD in F_Rat;
      end;
    end;
    then A1: Minor(M,i,j) in F_Rat by NAT_D:34;
    power(F_Real).(-1_F_Real,i+j) in F_Rat by LmSign1D;
    hence thesis by A1,RAT_1:def 2;
  end;
