
theorem LmSign1A:
  for n being Nat, M being Matrix of n, F_Real
  for H being Matrix of n, F_Rat
  st M = H
  holds Det M = Det H
  proof
    defpred P[Nat] means
    for M being Matrix of $1, F_Real
    for H being Matrix of $1, F_Rat st M = H holds Det M = Det H;
    P0: P[0]
    proof
      let M be Matrix of 0, F_Real;
      let H be Matrix of 0, F_Rat;
      assume M = H;
      Det M = 1.F_Real by MATRIXR2:41
      .= 1.F_Rat
      .= Det H by MATRIXR2:41;
      hence thesis;
    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;
      let H be Matrix of n+1, F_Rat;
      assume AS1: M = H;
      reconsider j = 1 as Nat;
      X0: 1 <= 1 & 1 <= n+1 by NAT_1:14;
      then JX:j in Seg (n+1);
      then
      X1: Det M = Sum LaplaceExpC(M,j) by LAPLACE:27;
      HX1: Det H = Sum LaplaceExpC(H,j) by JX,LAPLACE:27;
      set L = LaplaceExpC(M,j);
      set I = LaplaceExpC(H,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;
      Y3: dom L = Seg len I by X2,FINSEQ_1:def 3,LAPLACE:def 8
      .= dom I by FINSEQ_1:def 3;
      for i being Nat st i in dom L holds L.i = I.i
      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;
        HX31: I.i = H*(i,j)*Cofactor(H,i,j) by Y3,X30,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) = H*(i,j) by AS1,ZMATRLIN:1;
        Y1: (n+1)-'1 = n by NAT_D:34;
        set DD= Delete(M,i,j);
        set EE= Delete(H,i,j);
        Det DD = Det EE
        proof
          per cases;
          suppose 0 < n;
            hence Det DD = Det EE by AS1,P10,X41,Y1,LmSign1E;
          end;
          suppose not 0 < n;
            then Y2: n = 0;
            then Det DD = 1.F_Real by Y1,MATRIXR2:41
            .= 1.F_Rat
            .= Det EE by Y2,MATRIXR2:41,Y1;
            hence thesis;
          end;
        end;
        hence L.i = I.i by X32,X31,HX31,ZMATRLIN43;
      end;
      then L = I by Y3;
      hence thesis by X1,HX1,ZMATRLIN42;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(P0,P1);
    hence thesis;
  end;
