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 Th53:
  AS is AffinPlane & not LIN a,b,c implies ex c9 st a,c // a9,c9 & b,c // b9,c9
proof
  assume that
A1: AS is AffinPlane and
A2: not LIN a,b,c;
  consider C such that
A3: b in C & c in C and
A4: C is being_line by Th11;
  consider N such that
A5: b9 in N and
A6: C // N by A4,AFF_1:49;
A7: N is being_line by A6,AFF_1:36;
  consider P such that
A8: a in P and
A9: c in P and
A10: P is being_line by Th11;
  consider M such that
A11: a9 in M and
A12: P // M by A10,AFF_1:49;
A13: not M // N
  proof
    assume M // N;
    then N // P by A12,AFF_1:44;
    then C // P by A6,AFF_1:44;
    then b in P by A3,A9,AFF_1:45;
    hence contradiction by A2,A8,A9,A10,AFF_1:21;
  end;
  M is being_line by A12,AFF_1:36;
  then consider c9 such that
A14: c9 in M and
A15: c9 in N by A1,A7,A13,AFF_1:58;
A16: b,c // b9,c9 by A3,A5,A6,A15,AFF_1:39;
  a,c // a9,c9 by A8,A9,A11,A12,A14,AFF_1:39;
  hence thesis by A16;
end;
