theorem Th19:
  A out a,b & A out b,c implies A out a,c
  proof
    assume that
A1: A out a,b and
A2: A out b,c;
    consider x be POINT of S such that
A3: between a,A,x and
A4: between b,A,x by A1;
    consider y be POINT of S such that
A5: between b,A,y and
A6: between c,A,y by A2;
    between x,A,b & between y,A,b by A4,A5,GTARSKI3:14;
    then between y,A,c & A out y,x by A6,GTARSKI3:14;
    then between x,A,c by Th14;
    then between c,A,x by GTARSKI3:14;
    hence thesis by A3;
  end;
