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 Th82:
  A is_plane & not a in A & not b in A &
  b in half-space3(A,a) implies half-space3(A,b) c= 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);
    b in {x where x is POINT of S: A out2 x,a} by A1,A2,A4,Def18;
    then consider t be POINT of S such that
A5: b = t and
A6: A out2 t,a;
    half-space3(A,b) c= half-space3(A,a)
    proof
      let x be object;
      assume x in half-space3(A,b);
      then x in {x where x is POINT of S: A out2 x,b} by A1,A3,Def18;
      then consider y be POINT of S such that
A7:   x = y and
A8:   A out2 y,b;
      A out2 y,a by A5,A6,A8,Th79;
      then x in {x where x is POINT of S:A out2 x,a} by A7;
      hence thesis by A1,A2,Def18;
    end;
    hence thesis;
  end;
