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 Th74:
  between2 a,A,c implies (between2 b,A,c iff A out2 a,b)
  proof
    assume
A1: between2 a,A,c;
    hence between2 b,A,c implies A out2 a,b;
    assume A out2 a,b;
    then consider d be POINT of S such that
A2: between2 a,A,d and
A3: between2 b,A,d;
    consider x be POINT of S such that
A4: x in A and
A5: between a,x,d by A2;
    consider y be POINT of S such that
A6: y in A and
A7: between b,y,d by A3;
    A is_plane by A2;
    then consider p,q,r be POINT of S such that
A8: not Collinear p,q,r and
A9: A = Plane(p,q,r);
    per cases;
    suppose
A10:  Collinear a,b,d;
A11:  a <> d by A5,A4,A1,GTARSKI1:def 10;
      per cases;
      suppose a = b;
        hence thesis by A1;
      end;
      suppose
A12:    a <> b;
        x = y
        proof
          assume
A13:      x <> y;
            W1: x in Line(a,b)
            proof
                d in {x where x is POINT of S:Collinear a,b,x} by A10;
                then T1:Line(a,b) = Line(a,d) by A11,A12,GTARSKI3:82;
                Collinear a,x,d by A5;
                then Collinear a,d,x by GTARSKI3:45;
              hence thesis by T1;
            end;
            y in Line(a,b)
            proof
                  T2:b <> d by A3,GTARSKI1:def 10;
                  Collinear b,a,d by A10,GTARSKI3:45;
                  then d in Line(b,a);
                then H1: Line(a,b) = Line(b,d) by A12,T2,GTARSKI3:82;
                Collinear b,y,d by A7;
                then Collinear b,d,y by GTARSKI3:45;
              hence thesis by H1;
            end;
          then Line(x,y) = Line(a,b) & Line(x,y) c= Plane(p,q,r)
            by W1,A4,A6,A13,A12,A8,A9,GTARSKI4:11,Th69;
          hence thesis by A2,A9,GTARSKI3:83;
        end;
        then between d,x,a & between d,x,b by A7,A5,GTARSKI3:14;
        then x out a,b by A2,A4,A3,GTARSKI3:57;
        hence thesis by A1,A4,Th73;
      end;
    end;
    suppose
A14:  not Collinear a,b,d;
      consider z be POINT of S such that
A15:  between x,z,b and
A16:  between y,z,a by A5,A7,GTARSKI1:def 11;
      x <> y
      proof
        assume x = y;
        then between d,x,a & between d,x,b by A5,A7,GTARSKI3:14;
        then Collinear d,a,b or Collinear d,b,a by A2,A4,GTARSKI3:56;
        hence contradiction by A14,GTARSKI3:45;
      end;
      then
A17:  Line(x,y) c= A by A4,A6,A8,A9,Th69;
        y <> z
        proof
          assume y = z;
          then Collinear x,y,b by A15;
          then b in {t where t is POINT of S:Collinear x,y,t};
          hence thesis by A3,A17;
        end;
      then
A18:  y out a,z by A6,A1,A16;
        G1: x <> z
        proof
          assume x = z;
          then Collinear y,x,a by A16;
          then a in Line(y,x);
          hence contradiction by A1,A17;
        end;
A19:  x out z,b by A4,A3,G1,A15;
      between2 z,A,c by A18,A1,A6,Th73;
      hence thesis by A19,A4,Th73;
    end;
  end;
