theorem Satz4p14:
  Collinear a,b,c & a,b equiv a9,b9 implies ex c9 being POINT of S st
  a,b,c cong a9,b9,c9
  proof
    assume
A1: Collinear a,b,c & a,b equiv a9,b9;
    then per cases;
    suppose
A2:   between a,b,c;
      consider c9 be POINT of S such that
A3:   between a9,b9,c9 & b9,c9 equiv b,c by GTARSKI1:def 8;
A4:   b,c equiv b9,c9 by A3,Satz2p2;
      then a,c equiv a9,c9 by A1,A3,A2,Satz2p11;
      hence thesis by A1,A4,GTARSKI1:def 3;
    end;
    suppose between c,a,b;
      then
A5:   between b,a,c by Satz3p2;
      consider c9 be POINT of S such that
A6:   between b9,a9,c9 & a9,c9 equiv a,c by GTARSKI1:def 8;
      b,a equiv a9,b9 by A1,Satz2p4;
      then
A7:   b,a equiv b9,a9 by Satz2p5;
      a,c equiv a9,c9 by A6,Satz2p2;
      then
      b,a,c cong b9,a9,c9 by A5,A6,A7,Satz2p11;
      hence thesis by Lm4p14p1;
    end;
    suppose between b,c,a;
      then ex y be POINT of S st between a9,y,b9 & a,c,b cong a9,y,b9
        by A1,Satz3p2,Satz4p5;
      hence thesis by Lm4p14p2;
    end;
  end;
