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 Th29:
  A is_line & p in A & Collinear a,b,p implies
  (A out a,b iff (p out a,b & not a in A))
  proof
    assume that
A1: A is_line and
A2: p in A and
A3: Collinear a,b,p;
    hereby
      assume
A4:   A out a,b;
      consider c be POINT of S such that
A5:   between a,A,c and
A6:   between b,A,c by A4;
A7:   Collinear a,p,b by A3,GTARSKI3:45;
      not between a,p,b
      proof
        assume between a,p,b;
        then between a,A,b by A2,A5,A6;
        hence thesis by Th15,A4;
      end;
      hence p out a,b by A7,GTARSKI3:73;
      thus not a in A by A5;
    end;
    assume that
A8: p out a,b and
A9: not a in A;
    set c = reflection(p,a);
    between b,A,c iff A out a,b by A1,A2,A8,A9,Th28,Th27;
    hence thesis by A1,A2,A8,A9,Th27;
  end;
