theorem
  A is_plane & not r in A implies A c= space3(A,r) & r in space3(A,r)
  proof
    assume that
A1: A is_plane and
A2: not r in A;
    ex r9 be POINT of S st between2 r,A,r9 &
      space3(A,r) = half-space3(A,r) \/ A \/ half-space3(A,r9) by A1,A2,Def20;
    then
A3: A c= half-space3(A,r) \/ A & half-space3(A,r) \/ A c= space3(A,r)
      by XBOOLE_1:7;
    r in half-space3(A,r) & half-space3(A,r) c= space3(A,r)
      by A1,A2,Th80,Th84;
    hence thesis by A3;
  end;
