reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct;
reserve a,b for POINT of S;
reserve A for Subset of S;
reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct;
reserve a,b,c,m,r,s for POINT of S;
reserve A for Subset of S;
reserve S         for non empty satisfying_Lower_Dimension_Axiom
                                satisfying_Tarski-model
                                TarskiGeometryStruct,
        a,b,c,d,m,p,q,r,s,x for POINT of S,
        A,A9,E              for Subset of S;

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;
