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 Th27:
  A is_line & p in A & p out a,b & not a in A implies
  between b,A,reflection(p,a)
  proof
    assume that
A1: A is_line and
A2: p in A and
A3: p out a,b and
A4: not a in A;
    set c = reflection(p,a);
A5: between a,p,c
    proof
      Middle a,p,c by GTARSKI3:def 13;
      hence thesis;
    end;
A6: c <> p
    proof
      assume c = p;
      then Middle a,p,p by GTARSKI3:def 13;
      hence contradiction by A2,A4,GTARSKI1:def 7;
    end;
A7: between b,p,c by A5,A3,A6,GTARSKI3:71;
      N1: not b in A
      proof
        assume b in A;
        then
A8:     Line (p,b) = A by A3,A1,A2,GTARSKI3:87;
        Collinear a,p,b by A3,GTARSKI3:73;
        then Collinear p,b,a;
        hence thesis by A8,A4;
      end;
      not c in A
      proof
        assume c in A;
        then
A9:     Line(p,c) = A by A2,A1,A6,GTARSKI3:87;
        Collinear p,c,a by A5;
        hence contradiction by A9,A4;
      end;
    hence thesis by N1,A1,A2,A7;
  end;
