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 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;
