theorem Th83:
  A is_plane & not a in A & not b in A &
  b in half-space3(A,a) implies half-space3(A,b) = half-space3(A,a)
  proof
    assume that
A1: A is_plane and
A2: not a in A and
A3: not b in A and
A4: b in half-space3(A,a);
    half-space3(A,a) c= half-space3(A,b)
    proof
      a in half-space3(A,b) by A1,A2,A3,A4,Th81;
      hence thesis by A1,A2,A3,Th82;
    end;
    hence thesis by A1,A2,A3,A4,Th82;
  end;
