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
  for M be Matrix of n,REAL
  holds
    M is invertible iff MXR2MXF M is invertible
  proof
    let M be Matrix of n,REAL;
    hereby
      assume M is invertible;
      then consider B be Matrix of n,REAL such that
      A1: (B * M = 1_Rmatrix n & M * B = 1_Rmatrix n) by MATRIXR2:def 5;
      thus MXR2MXF M is invertible by A1,MATRIX_6:def 2,def 3;
    end;
    assume MXR2MXF M is invertible;
    then consider M2 be Matrix of n,F_Real such that
    A2: MXR2MXF M is_reverse_of M2 by MATRIX_6:def 3;
    set B = MXF2MXR M2;
    A3: M*B = 1_Rmatrix n by A2,MATRIX_6:def 2;
    B*M = 1_Rmatrix n by A2,MATRIX_6:def 2;
    hence M is invertible by A3,MATRIXR2:def 5;
  end;
