theorem Th48:
  a,b '||' c,d implies
  (a,c '||' b,d or ex x st a,c,x are_collinear & b,d,x are_collinear )
proof
  assume
A1: a,b '||' c,d;
  assume
A2: not a,c '||' b,d;
A3: now
    consider z such that
A4: a,b '||' c,z and
A5: a,c '||' b,z by DIRAF:26;
    assume
A6: a<>b;
A7: now
      c,d '||' c,z by A1,A6,A4,DIRAF:23;
      then c,d,z are_collinear by DIRAF:def 5;
      then d,c,z are_collinear by DIRAF:30;
      then d,c '||' d,z by DIRAF:def 5;
      then z,d '||' d,c by DIRAF:22;
      then consider t such that
A8:   b,d '||' d,t and
A9:   b,z '||' c,t by A2,A5,DIRAF:27;
      assume b<>z;
      then a,c '||' c,t by A5,A9,DIRAF:23;
      then c,a '||' c,t by DIRAF:22;
      then c,a,t are_collinear by DIRAF:def 5;
      then
A10:  a,c,t are_collinear by DIRAF:30;
      d,b '||' d,t by A8,DIRAF:22;
      then d,b,t are_collinear by DIRAF:def 5;
      then b,d,t are_collinear by DIRAF:30;
      hence thesis by A10;
    end;
    now
      assume b=z;
      then b,a '||' b,c by A4,DIRAF:22;
      then b,a,c are_collinear by DIRAF:def 5;
      then
A11:  a,c,b are_collinear by DIRAF:30;
      b,d,b are_collinear by DIRAF:31;
      hence thesis by A11;
    end;
    hence thesis by A7;
  end;
  now
    assume a=b;
    then a,c,a are_collinear & b,d,a are_collinear by DIRAF:31;
    hence thesis;
  end;
  hence thesis by A3;
end;
