reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;

theorem Th52:
  for n being Nat for N being Matrix of n,REAL st
  N is invertible holds N@ is invertible & Inv(N@) = (Inv(N))@
  proof
    let n be Nat;
    let N be Matrix of n,REAL;
    assume
A1: N is invertible;
    then (N * Inv(N))@ = (1_Rmatrix(n))@ by MATRIXR2:def 6
                      .= 1_Rmatrix(n) by MATRIXR2:64; then
A2: (Inv(N)@) * N@ = 1_Rmatrix(n) by MATRIXR2:30;
    (Inv(N) * N)@ = (1_Rmatrix(n))@ by A1,MATRIXR2:def 6
                 .= 1_Rmatrix(n) by MATRIXR2:64; then
A3: (N@ * Inv(N)@) = 1_Rmatrix(n) by MATRIXR2:30; then
    N@ is invertible by A2,MATRIXR2:def 5;
    hence thesis by A2,A3,MATRIXR2:def 6;
  end;
