theorem
  for a being non zero Real
  for N being Matrix of 3,F_Real st N = symmetric_3(a,a,-a,0,0,0) holds
  N is invertible
  proof
    let a be non zero Real;
    let N be Matrix of 3,F_Real;
    assume
A1: N = symmetric_3(a,a,-a,0,0,0);
    symmetric_3(a,a,-a,0,0,0) * symmetric_3(1/a,1/a,-1/a,0,0,0)
      = 1.(F_Real,3) &
    symmetric_3(1/a,1/a,-1/a,0,0,0) * symmetric_3(a,a,-a,0,0,0)
      = 1.(F_Real,3) by Th41,Th42;
    hence thesis by A1,MATRIX_6:def 2,def 3;
  end;
