theorem Th14:
  not p,a,b are_collinear & p,a '||' b,c & p,b '||' a,c implies
   p,a // b,c & p,b // a,c
proof
  assume that
A1: not p,a,b are_collinear and
A2: p,a '||' b,c and
A3: p,b '||' a,c;
  consider d such that
A4: p,a // b,d and
A5: p,b // a,d and
  a<>d by ANALOAF:def 5;
A6: p,b '||' a,d by A5,DIRAF:def 4;
  p,a '||' b,d by A4,DIRAF:def 4;
  hence thesis by A1,A2,A3,A4,A5,A6,Th5;
end;
