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 Th33:
  X c= Y & ( X is being_line & Y is being_line or X is being_plane
  & Y is being_plane ) implies X=Y
proof
  assume that
A1: X c= Y and
A2: X is being_line & Y is being_line or X is being_plane & Y is being_plane;
A3: now
    assume that
A4: X is being_plane and
A5: Y is being_plane;
    consider K,P such that
A6: K is being_line and
A7: P is being_line and
A8: not K // P and
A9: X=Plane(K,P) by A4;
    consider a,b such that
A10: a in P and
    b in P and
    a<>b by A7,AFF_1:19;
    set M=a*K;
A11: K // M by A6,Def3;
A12: P c= X by A6,A9,Th14;
    then
A13: P c= Y by A1;
A14: M is being_line by A6,Th27;
    a in M & P c= Plane(K,P) by A6,Def3,Th14;
    then
A15: M c= X by A9,A10,A11,Lm4;
    then M c= Y by A1;
    hence thesis by A4,A5,A7,A8,A11,A14,A12,A13,A15,Th26;
  end;
  now
    assume that
A16: X is being_line and
A17: Y is being_line;
    consider a,b such that
A18: a<>b and
A19: X=Line(a,b) by A16,AFF_1:def 3;
    a in X & b in X by A19,AFF_1:15;
    hence thesis by A1,A17,A18,A19,AFF_1:57;
  end;
  hence thesis by A2,A3;
end;
