theorem Th84:
  A is_plane & not r in A implies half-space3(A,r) c= space3(A,r)
  proof
    assume A is_plane & not r in A;
    then consider r9 be POINT of S such that
    between2 r,A,r9 and
A1: space3(A,r) = half-space3(A,r) \/ A \/ half-space3(A,r9) by Def20;
    half-space3(A,r) c= half-space3(A,r) \/ A &
      half-space3(A,r) \/ A c= half-space3(A,r) \/ A \/ half-space3(A,r9)
      by XBOOLE_1:7;
    hence thesis by A1;
  end;
