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 Th81:
  A is_plane & not a in A & not b in A & b in half-space3(A,a) implies
  a in half-space3(A,b)
  proof
    assume
A1: A is_plane & not a in A & not b in A & b in half-space3(A,a);
    then b in {x where x is POINT of S:A out2 x,a} by Def18;
    then ex x be POINT of S st x = b & A out2 x,a;
    then A out2 a,b;
    then a in {x where x is POINT of S:A out2 x,b};
    hence thesis by A1,Def18;
  end;
