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

theorem
  for A being Matrix of 3,REAL holds 
    Det A = (A*(1,1))*(A*(2,2))*(A*(3,3))
          - (A*(1,3))*(A*(2,2))*(A*(3,1)) 
          - (A*(1,1))*(A*(2,3))*(A*(3,2))
          + (A*(1,2))*(A*(2,3))*(A*(3,1)) 
          - (A*(1,2))*(A*(2,1))*(A*(3,3)) 
          + (A*(1,3))*(A*(2,1))*(A*(3,2))
proof
  let A be Matrix of 3,REAL;
  reconsider N=MXR2MXF A as Matrix of 3,F_Real;
  reconsider N2=<* <* N*(1,1),N*(1,2), N*(1,3) *>, <* N*(2,1),N*(2,2), N*(2,3)
  *>, <* N*(3,1),N*(3,2), N*(3,3) *> *> as Matrix of 3,F_Real by Th35;
  Det A = Det N2 by Th37
    .= (N*(1,1))*(N*(2,2))*(N*(3,3))-(N*(1,3))*(N*(2,2))*(N*(3,1)) -(N*(1,1)
)*(N*(2,3))*(N*(3,2))+(N*(1,2))*(N*(2,3))*(N*(3,1)) -(N*(1,2))*(N*(2,1))*(N*(3,
  3))+(N*(1,3))*(N*(2,1))*(N*(3,2)) by MATRIX_9:46;
  hence thesis;
end;
