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
  between a,c,p & between b,q,c implies ex x st between a,x,b & between p,q,x
  proof
    assume that
A1: between a,c,p and
A2: between b,q,c;
    per cases;
    suppose
A3:   p = q;
      take b;
      thus thesis by A3,GTARSKI3:13,14;
    end;
    suppose
A4:   p <> q;
      per cases;
      suppose
A5:     c = q;
        take a;
        thus thesis by A1,A5,GTARSKI3:13,14;
      end;
      suppose
A6:     c <> q;
A7:     between p,c,a by A1,GTARSKI3:14;
        per cases;
        suppose
A8:       Collinear p,q,c;
          per cases;
          suppose
A9:         between p,q,c;
            take a;
            thus thesis by A7,A9,GTARSKI3:15,18;
          end;
          suppose not between p,q,c;
            then
A10:        q out p,c by A8,GTARSKI3:73;
            then per cases;
            suppose between q,p,c;
              then
A11:          between c,q,b & between c,p,q by A2,GTARSKI3:14;
              take b;
              thus thesis by A11,GTARSKI3:13,18;
            end;
            suppose between q,c,p;
              then
A12:          between c,q,b & between p,c,q by A2,GTARSKI3:14;
              take b;
              thus thesis by A10,A12,GTARSKI3:13,19;
            end;
          end;
        end;
        suppose
A13:      not Collinear p,q,c;
          set A = Line(p,q);
A14:      Line(p,q) <> Line(c,q)
          proof
            assume Line(p,q) = Line(c,q);
            then c in Line(p,q) by GTARSKI3:83;
            then ex x be POINT of S st c = x & Collinear p,q,x;
            hence contradiction by A13;
          end;
          per cases;
          suppose
A15:        b in A;
            Collinear b,q,c by A2;
            then Collinear c,q,b by GTARSKI3:45;
            then
A16:        b in Line(c,q);
A17:        q in A & q in Line(c,q) by GTARSKI3:83;
            take b;
            b = q
            proof
              assume
A18:          b <> q;
                A is_line by A4;
                then G1: Line(b,q) = A by A15,A17,A18,GTARSKI3:87;
                Line(c,q) is_line by A6;
              hence contradiction by G1,A14,A16,A17,A18,GTARSKI3:87;
            end;
            hence thesis by GTARSKI3:13;
          end;
          suppose
A19:        not b in A;
                R1: A is_line by A4;
                R2: not c in A
                proof
                  assume
A20:              c in A;
                  q in A & A is_line by A4,GTARSKI3:83;
                  then
A21:              Line(c,q) = A by A20,A6,GTARSKI3:87;
                  Collinear c,q,b by A2,GTARSKI3:14;
                  hence contradiction by A19,A21;
                end;
                between b,q,c & q in A by A2,GTARSKI3:83;
              then T2: between c,A,b by R1,R2,A19,GTARSKI3:14;
                O1:p <> c
                proof
                  assume p = c;
                  then Collinear p,c,q by GTARSKI3:46;
                  hence contradiction by A13,GTARSKI3:45;
                end;
                p <> a
                proof
                  assume p = a;
                  then Collinear b,q,p by A2,A1,GTARSKI1:def 10;
                  then Collinear p,q,b by GTARSKI3:45;
                  hence contradiction by A19;
                end;
              then p out c,a by O1,A1,GTARSKI3:14;
            then between a,A,b by GTARSKI3:83,T2,Th12;
            then consider x be POINT of S such that
A22:        x in A and
A23:        between a,x,b;
              P1: p <> x
              proof
                assume p = x;
                then between c,p,b &between c,q,b by A23,A1,A2,GTARSKI3:14,18;
                then Collinear c,p,q or Collinear c,q,p by GTARSKI3:58;
                hence contradiction by A13,GTARSKI3:45;
              end;
               A is_line & p in A by A4,GTARSKI3:83;
            then
A24:        Line(p,x) = A by P1,A22,GTARSKI3:87;
            take x;
              between b,x,a & between p,c,a by A1,A23,GTARSKI3:14;
              then consider t be POINT of S such that
A25:          between x,t,p and
A26:          between c,t,b by GTARSKI1:def 11;
A27:           Collinear p,x,t by A25;
                Collinear c,t,b by A26;
                then Collinear c,b,t by GTARSKI3:45;
                then
A28:            t in Line(c,b);
                  R1: c <> b by A6,A2,GTARSKI1:def 10;
                  Collinear c,b,q by A2;
                  then q in Line(c,b);
                then t in Line(c,q) & t in A & q in Line(c,q) & q in A &
                  Line(c,q) is_line & A is_line
                  by A4,A27,A28,A24,GTARSKI3:82,83,R1,A6;
                then t=q by A14,GTARSKI3:89;
            hence thesis by A23,A25,GTARSKI3:14;
          end;
        end;
      end;
    end;
  end;
