reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;
reserve A,C,D,K for Subset of AS;
reserve AP for AffinPlane;
reserve a,b,c,d,x,p,q for Element of AP;
reserve A,C for Subset of AP;

theorem Th57:
  A is being_line & C is being_line & not A // C implies
    ex x st x in A & x in C
proof
  assume that
A1: A is being_line and
A2: C is being_line and
A3: not A // C;
  consider a,b such that
A4: a<>b and
A5: A=Line(a,b) by A1;
  consider c,d such that
A6: c <>d and
A7: C=Line(c,d) by A2;
  not a,b // c,d by A3,A4,A5,A6,A7,Th36;
  then consider x such that
A8: a,b // a,x and
A9: c,d // c,x by DIRAF:46;
  LIN c,d,x by A9;
  then
A10: x in C by A7,Def2;
  LIN a,b,x by A8;
  then x in A by A5,Def2;
  hence thesis by A10;
end;
