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 Th28:
  A is_line & p in A & not a in A implies between a,A,reflection(p,a)
  proof
    assume that
A1: A is_line and
A2: p in A and
A3: not a in A;
    set c = reflection(p,a);
A4: between a,p,c
    proof
      Middle a,p,c by GTARSKI3:def 13;
      hence thesis;
    end;
A5: c <> p
    proof
      assume c = p;
      then Middle a,p,p by GTARSKI3:def 13;
      hence contradiction by A2,A3,GTARSKI1:def 7;
    end;
      not c in A
      proof
        assume c in A;
        then
A6:     Line(p,c) = A by A2,A1,A5,GTARSKI3:87;
        Collinear p,c,a by A4;
        hence contradiction by A6,A3;
      end;
    hence thesis by A1,A2,A4,A3;
  end;
