theorem
  not Collinear a,b,c & b <> d & a,b equiv c,d & b,c equiv d,a &
  Collinear a,p,c & Collinear b,p,d implies Middle a,p,c & Middle b,p,d
  proof
    assume that
A1: not Collinear a,b,c and
A2: b <> d and
A3: a,b equiv c,d and
A4: b,c equiv d,a and
A5: Collinear a,p,c and
A6: Collinear b,p,d;
A7: Collinear b,d,p by A6,Satz3p2;
    then consider p9 being POINT of S such that
A8: b,d,p cong d,b,p9 by GTARSKI1:def 5,Satz4p14;
A9: Collinear d,b,p9 by A7,A8,Satz4p13;
    now
      thus Collinear b,d,p by A6,Satz3p2;
      thus b,d,p cong d,b,p9 by A8;
      a,b equiv d,c by A3,Satz2p5;
      hence b,a equiv d,c by Satz2p4;
      thus d,a equiv b,c by A4,Satz2p2;
    end;
    then
A10: b,d,p,a FS d,b,p9,c;
    then
    p,a equiv c,p9 by A2,Satz4p16,Satz2p5;
    then
A11: a,p equiv c,p9 by Satz2p4;
A12: a,c equiv c,a by GTARSKI1:def 5;
    now
      thus Collinear b,d,p by A6,Satz3p2;
      thus b,d,p cong d,b,p9 by A8;
      thus b,c equiv d,a by A4;
      a,b equiv d,c by A3,Satz2p5;
      then b,a equiv d,c by Satz2p4;
      hence d,c equiv b,a by Satz2p2;
    end;
    then b,d,p,c FS d,b,p9,a;
    then p,c equiv p9,a by A2,Satz4p16;
    then
A13: Collinear c,p9,a by A5,Satz4p13,A11,A12,GTARSKI1:def 3;
A14: a <> c by A1,Satz3p1;
A15: Line(a,c) <> Line(b,d)
    proof
      assume Line(a,c) = Line(b,d);
      then b in Line(a,c) by Satz6p17;
      then ex x be POINT of S st b = x & Collinear a,c,x;
      hence contradiction by A1,Satz3p2;
    end;
    now
      thus Line(a,c) <> Line(b,d) by A15;
      thus Line(a,c) is_line by A14;
      thus Line(b,d) is_line by A2;
      Collinear a,c,p9 by A13;
      hence p9 in Line(a,c);
      Collinear b,d,p9 by A9,Satz3p2;
      hence p9 in Line(b,d);
      Collinear a,c,p by A5,Satz3p2;
      hence p in Line(a,c);
      Collinear b,d,p by A6,Satz3p2;
      hence p in Line(b,d);
    end;
    then
A16: p9 = p by Satz6p19;
    now
      thus Middle a,p,c by A14,Satz7p20,A10,A2,Satz4p16,A16,A5;
      b,p equiv p,d by A16,A8,Satz2p5;
      hence Middle b,p,d by A2,Satz7p20,A6,Satz2p4;
    end;
    hence thesis;
  end;
