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
  A out a,b & between a,c,b implies A out c,a
  proof
    assume that
A1: A out a,b and
A2: between a,c,b;
    consider d be POINT of S such that
A3: between a,A,d and
A4: between b,A,d by A1;
    consider x be POINT of S such that
A5: x in A and
A6: between a,x,d by A3;
    consider y be POINT of S such that
A7: y in A and
A8: between b,y,d by A4;
    consider t be POINT of S such that
A9: between c,t,d and
A10: between x,t,y by A2,A6,A8,GTARSKI3:40;
A11: not c in A
    proof
      assume c in A;
      then between a,A,b by A2,A3,A4;
      hence contradiction by A1,Th15;
    end;
      between c,A,d
      proof
        per cases;
        suppose x = y;
          then t = x by A10,GTARSKI1:def 10;
          hence between c,A,d by A5,A11,A3,A9;
        end;
        suppose A12: x <> y;
              G1: Line(x,y) = A by A12,A5,A7,A1,Th24,GTARSKI3:87;
              Collinear x,y,t by A10,GTARSKI3:14;
            then t in A by G1;
          hence between c,A,d by A11,A3,A9;
        end;
      end;
      hence thesis by A3;
  end;
