reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);

theorem
  for n being Nat for M being Matrix of n,REAL
  for N being Matrix of n,F_Real st M = N holds
  M is invertible iff N is invertible
  proof
    let n be Nat;
    let M be Matrix of n,REAL;
    let N be Matrix of n,F_Real;
    assume
A1: M = N;
A2: 1.(F_Real,n) = MXF2MXR 1.(F_Real,n) by MATRIXR1:def 2
                .= 1_Rmatrix(n) by MATRIXR2:def 2;
    hereby
      assume M is invertible;
      then consider B be Matrix of n,REAL such that
A3:   B * M = 1_Rmatrix(n) and
A4:   M * B = 1_Rmatrix(n) by MATRIXR2:def 5;
      reconsider C = B as Matrix of n,F_Real;

A5:   B * M = C * N by A1,Th15;
      C is_reverse_of N
      proof
        C * N = N * C by A1,A3,A4,A5,Th15;
        hence thesis by A2,A3,A1,Th15,MATRIX_6:def 2;
      end;
      hence N is invertible by MATRIX_6:def 3;
    end;
    assume N is invertible;
    then consider N2 be Matrix of n,F_Real such that
A6: N is_reverse_of N2 by MATRIX_6:def 3;
    reconsider M2 = N2 as Matrix of n,REAL;
    now
      take M2;
A7:   M2 * M = N2 * N by A1,Th15
            .= 1.(F_Real,n) by A6,MATRIX_6:def 2;
      hence M2 * M = 1_Rmatrix(n) by A2;
      M2 * M = N2 * N by A1,Th15
            .= N * N2 by A6,MATRIX_6:def 2
            .= M * M2 by A1,Th15;
      hence M * M2 = 1_Rmatrix(n) by A7,A2;
    end;
    hence M is invertible by MATRIXR2:def 5;
  end;
