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 Th73:
  between2 a,A,c & r in A implies
  (for b st r out a,b holds between2 b,A,c)
  proof
    assume that
A1: between2 a,A,c and
A2: r in A;
    consider t be POINT of S such that
A3: t in A and
A4: between a,t,c by A1;
    A is_plane by A1;
    then consider p9,q9,r9 be POINT of S such that
A5: not Collinear p9,q9,r9 and
A6: A = Plane(p9,q9,r9);
    hereby
      let b;
      assume
A7:   r out a,b;
A8:   not b in A
      proof
        assume b in A; then
T1:     Line(r,b) c= A by A2,A7,A5,A6,Th69;
        Collinear a,r,b by A7,GTARSKI3:73;
        then Collinear r,b,a;
        then a in Line(r,b);
        hence thesis by T1,A1;
      end;
      thus between2 b,A,c
      proof
        per cases;
        suppose
A10:      r = t;
            between b,t,c
            proof
              per cases by A7;
              suppose between r,b,a;
                then between a,b,t & between a,t,c by A4,A10,GTARSKI3:14;
                hence thesis by GTARSKI3:18;
              end;
              suppose between r,a,b;
                then between b,a,t & between a,t,c & a <> t
                  by A10,A4,A1,A3,GTARSKI3:14;
                hence thesis by GTARSKI3:19;
              end;
          end;
          hence between2 b,A,c by A1,A3,A8;
        end;
        suppose
A11:      r <> t;
          set L = Line(r,t);
A12:      Line(r,t) c= A by A2,A11,A3,A5,A6,Th69;
            L is_line & not a in L & not c in L & ex t being POINT of S st
              t in L & between a,t,c by A1,A4,A11,A12,GTARSKI3:83;
            then between a,L,c;
          then between b,L,c by A7,Th12,GTARSKI3:83;
          hence between2 b,A,c by A1,A8,A12;
        end;
      end;
    end;
  end;
