theorem
  c,a equiv c,b implies ex x being POINT of S st Middle a,x,b
  proof
    assume
A1: c,a equiv c,b;
    per cases;
    suppose Collinear a,b,c;
      then Collinear a,c,b by Satz3p2;
      then per cases by A1,Satz7p20;
      suppose
A2:     a = b;
        take a;
        thus thesis by A2,Satz3p1,Satz2p1;
      end;
      suppose Middle a,c,b;
        hence thesis;
      end;
    end;
    suppose
A3:   not Collinear a,b,c;
      consider p such that
A4:   between c,a,p and
A5:   a <> p by Satz3p14;
      consider q such that
A6:   between c,b,q and
A7:   b,q equiv a,p by GTARSKI1:def 8;
      between p,a,c & between q,b,c by A4,A6,Satz3p2;
      then consider r such that
A8:   between a,r,q and
A9:   between b,r,p by GTARSKI1:def 11;
      consider x be POINT of S such that
A10:  between a,x,b and
A11:  between r,x,c by A4,A9,GTARSKI1:def 11;
      take x;
      x,a equiv x,b
      proof
A12:    r,a equiv r,b
        proof
A13:      c,a,p,b AFS c,b,q,a
            by A4,A6,A1,A7,GTARSKI1:def 5,Satz2p2;
          c <> a by A3,Satz3p1; then
A15:      p,b equiv a,q by Satz2p5,A13,Axiom5AFS;
          thesis
          proof
            consider r9 be POINT of S such that
A19:        between a,r9,q and
A20:        b,r,p cong a,r9,q by A15,Satz2p4,A9,Satz4p5;
A21:        now
              b,q equiv p,a by A7,Satz2p5;
              then q,b equiv p,a by Satz2p4;
              hence b,r,p,a IFS a,r9,q,b
                by A9,A19,A20,Satz2p2,GTARSKI1:def 5;
              thus b,r,p,q IFS a,r9,q,p by A9,A19,A20,GTARSKI1:def 5,A7;
            end;
            then r,a equiv b,r9 by Satz2p5,Satz4p2; then
A23:        a,r,q cong b,r9,p by A21,Satz2p4,Satz2p2,Satz4p2;
            now
              thus Collinear a,r,q by A8;
              hence Collinear b,r9,p by A23,Satz4p13;
            end;
            then Collinear a,q,r & Collinear b,p,r9 by Satz3p2;
            then
A24:        r in Line(a,q) & r9 in Line(b,p);
A25:        r9 in Line(a,q) & r in Line(b,p)
            proof
              Collinear a,q,r9 by A19,Satz3p2;
              hence r9 in Line(a,q);
              Collinear b,p,r by A9,Satz3p2;
              hence r in Line(b,p);
            end;
A26:        a = q iff b = p by A15,Satz2p2,GTARSKI1:def 7;
            r = r9
            proof
              per cases;
              suppose
A27:            a = q or b = p;
                then
A28:            a = r by A8,A26,GTARSKI1:def 10;
                between a,r9,a by A19,A27,A15,Satz2p2,GTARSKI1:def 7;
                hence thesis by A28,GTARSKI1:def 10;
              end;
              suppose
A29:            a <> q & b <> p;
                assume
A30:            r <> r9;
                reconsider A = Line(a,q), B = Line(b,p) as Subset of S;
A31:            A is_line & B is_line by A29;
                then
A32:            A = B by A30,A24,A25,Satz6p19;
A33:            a <> b by A3,Satz3p1;
A34:            A = Line(a,b)
                proof
                  b in A & a in A & A is_line by A29,A32,Satz6p17;
                  hence thesis by A33,Satz6p18;
                end;
A35:            Line(a,p) = A
                proof
A36:              a in A by Satz6p17;
                  p in B by Satz6p17;
                  then p in A & A is_line by A31,A30,A24,A25,Satz6p19;
                  hence Line(a,p) = A by A36,A5,Satz6p18;
                end;
                c in A
                proof
                  Collinear a,p,c by A4;
                  hence thesis by A35;
                end;
                then ex y be POINT of S st c = y & Collinear a,b,y by A34;
                hence contradiction by A3;
              end;
            end;
            hence thesis by A21,Satz4p2;
          end;
          hence thesis;
        end;
        per cases;
        suppose r = c;
          then r = x by A11,GTARSKI1:def 10;
          hence thesis by A12;
        end;
        suppose
A37:      r <> c;
          Collinear c,r,x & c,a equiv c,b & r,a equiv r,b by A11,A1,A12;
          hence thesis by A37,Satz4p17;
        end;
      end;
      hence thesis by A10;
    end;
  end;
