
theorem LmGM11:
  for n being Nat, M being Matrix of n, F_Real st M is Matrix of n, F_Rat
  holds Det M in F_Rat
  proof
    defpred P[Nat] means
    for M being Matrix of $1, F_Real st M is Matrix of $1, F_Rat
    holds Det M in F_Rat;
    P0: P[0]
    proof
      let M be Matrix of 0, F_Real;
      assume M is Matrix of 0, F_Rat;
      Det M = 1.F_Real by MATRIXR2:41
      .= 1;
      hence Det M in F_Rat;
    end;
    P1: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume P10: P[n];
      let M be Matrix of n+1, F_Real;
      assume AS1: M is Matrix of n+1, F_Rat;
      reconsider j = 1 as Nat;
      X0: 1 <= 1 & 1 <= n+1 by NAT_1:14;
      then j in Seg (n+1);
      then
      X1: Det M = Sum LaplaceExpC(M,j) by LAPLACE:27;
      set L = LaplaceExpC(M,j);
      X2: len L = n+1
      & for i being Nat st i in dom L holds
      L.i = M*(i,j)*Cofactor(M,i,j) by LAPLACE:def 8;
      for i being Nat st i in dom L holds L.i in F_Rat
      proof
        let i be Nat;
        assume X30:i in dom L; then
        X31: L.i = M*(i,j)*Cofactor(M,i,j) by LAPLACE:def 8;
        i in Seg (n+1) & j in Seg (n+1) by X0,X2,X30,FINSEQ_1:def 3;
        then [i,j] in [:Seg (n+1),Seg (n+1):] by ZFMISC_1:87;
        then
        X41: [i, j] in Indices M by MATRIX_0:24;
        then
        X32: M*(i,j) is Element of F_Rat by AS1,ZMATRLIN:41;
        (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,AS1,X41;
            hence Det DD in F_Rat by P10;
          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 L.i in F_Rat by A1,X32,X31,RAT_1:def 2;
      end;
      hence thesis by X1,LmSign1C;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(P0,P1);
    hence thesis;
  end;
