theorem Th67:
  a,b _|_ c,d implies ex p st LIN a,b,p & LIN c,d,p
proof
  reconsider a9=a,b9=b,c9=c,d9=d as Element of the AffinStruct of POS;
  assume
A1: a,b _|_ c,d;
A2: now
    set M = Line(a,b),N = Line(c,d);
    assume a<>b & c <>d;
    then M _|_ N by A1,Th45;
    then consider p such that
A3: p in M & p in N by Th66;
    LIN a,b,p & LIN c,d,p by A3,Def10;
    hence thesis;
  end;
  LIN a9,b9,a9 by AFF_1:7;
  then
A4: LIN a,b,a by Th40;
A5: now
    assume c =d;
    then c,d // c,a by Th58;
    then LIN c,d,a;
    hence thesis by A4;
  end;
  LIN c9,d9,c9 by AFF_1:7;
  then
A6: LIN c,d,c by Th40;
  now
    assume a=b;
    then a,b // a,c by Th58;
    then LIN a,b,c;
    hence thesis by A6;
  end;
  hence thesis by A5,A2;
end;
