reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;
reserve n,i,j for Nat;
reserve n for Nat;

theorem
  for A being Matrix of n,REAL st A is invertible holds Det Inv A = (Det A)"
proof
  let A be Matrix of n,REAL;
  assume A is invertible;
  then A*(Inv A)=1_Rmatrix(n) by Def6;
  then Det (A*(Inv A))=1 by Th72;
  then
A1: Det(A)*Det (Inv A)=1 by Th46;
  per cases;
  suppose
    Det A <> 0;
    hence thesis by A1,XCMPLX_0:def 7;
  end;
  suppose
    Det A=0;
    hence thesis by A1;
  end;
end;
