theorem Th80:
  A is_plane & not a in A implies a in half-space3(A,a)
  proof
    assume that
A1: A is_plane and
A2: not a in A;
A3: half-space3(A,a) = {x where x is POINT of S: A out2 x,a} by A1,A2,Def18;
    A out2 a,a by Th77,A1,A2;
    hence thesis by A3;
  end;
