reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th33:
  for M be Matrix of n,F_Real
  for N be Matrix of n,REAL
    st N = MXF2MXR M
  holds
    M is invertible iff N is invertible
  proof
    let M be Matrix of n,F_Real;
    let N be Matrix of n,REAL;
    assume
    A1: N = MXF2MXR M;

    hereby
      assume M is invertible; then
      consider M2 be Matrix of n,F_Real such that
      A2: M is_reverse_of M2 by MATRIX_6:def 3;
      set B = MXF2MXR M2;
      A3: B*N = 1_Rmatrix n by A1,A2,MATRIX_6:def 2;
      N*B = 1_Rmatrix n by A1,A2,MATRIX_6:def 2;
      hence N is invertible by MATRIXR2:def 5,A3;
    end;

    assume N is invertible; then
    ex B be Matrix of n,REAL st
    (B * N = 1_Rmatrix n & N * B = 1_Rmatrix n) by MATRIXR2:def 5;
    hence M is invertible by A1,MATRIX_6:def 2,def 3;
  end;
