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 Th16:
  A is_line & not a in A implies ex c st between a,A,c
  proof
    assume that
A1: A is_line and
A2: not a in A;
    consider p,q such that
A3: p <> q and
A4: A = Line(p,q) by A1;
A5: p in A & q in A by A4,GTARSKI3:83;
    set c = reflection(p,a);
A6: p <> c
    proof
      assume p = c;
      then Middle a,c,c by GTARSKI3:def 13;
      hence contradiction by A2,A5,GTARSKI3:104,GTARSKI1:def 7;
    end;
    take c;
A7:   Middle a,p,c by GTARSKI3:def 13;
      Collinear p,c,a by A7;
      then a in Line(p,c);
    hence thesis by A4,A1,A7,GTARSKI3:82,83,A2,A3,A6;
  end;
