reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem Th29:
  X is being_plane & a in X & b in X & c in X & a,b // c,d & a<>b
  implies d in X
proof
  assume that
A1: X is being_plane & a in X & b in X & c in X and
A2: a,b // c,d and
A3: a<>b;
  set M=Line(a,b), N=c*M;
A4: M is being_line by A3,AFF_1:def 3;
  then
A5: N c= X by A1,A3,Th19,Th28;
A6: a in M & b in M by AFF_1:15;
  c in N & M // N by A4,Def3;
  then d in N by A2,A3,A6,Th7;
  hence thesis by A5;
end;
