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 Th36:
  for a,A st A is being_line ex X st a in X & A c= X & X is being_plane
proof
  let a,A;
  assume
A1: A is being_line;
  then consider p,q such that
A2: p in A and
  q in A and
  p<>q by AFF_1:19;
A3: now
    consider b such that
A4: not b in A by A1,Th12;
    consider P such that
A5: a in P & b in P and
A6: P is being_line by Th11;
    set X=Plane(P,A);
A7: A c= X by A6,Th14;
    assume
A8: a in A;
    then not P // A by A4,A5,AFF_1:45;
    then X is being_plane by A1,A6;
    hence thesis by A8,A7;
  end;
  now
    consider P such that
A9: a in P and
A10: p in P and
A11: P is being_line by Th11;
    set X=Plane(P,A);
    A c= X by A11,Th14;
    then
A12: P c= X by A2,A10,A11,Lm4,AFF_1:41;
    assume not a in A;
    then not P // A by A2,A9,A10,AFF_1:45;
    then X is being_plane by A1,A11;
    hence thesis by A9,A11,A12,Th14;
  end;
  hence thesis by A3;
end;
