reserve X for OrtAfPl;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1,c2,c3,d,d1,d2,d3,d4,e1,e2 for
  Element of X;
reserve a29,a39,b29,x9 for Element of the AffinStruct of X;
reserve A,K,M,N for Subset of X;
reserve A9,K9 for Subset of the AffinStruct of X;

theorem Th3:
  for A,a st A is being_line ex K st a in K & A _|_ K
proof
  let A,a;
  assume A is being_line;
  then consider b,c such that
A1: b<>c and
A2: A = Line(b,c) by ANALMETR:def 12;
  consider d such that
A3: b,c _|_ a,d and
A4: a<>d by ANALMETR:39;
  reconsider a9=a,d9=d as Element of the AffinStruct of X;
  take K = Line(a,d);
  K = Line(a9,d9) by ANALMETR:41;
  hence a in K by AFF_1:15;
  thus thesis by A1,A2,A3,A4,ANALMETR:45;
end;
