
theorem Th5:  :: belongs to PENCIL_1
for X being set, v being object st 3 c= card X
  ex v1, v2 being object st v1 in X & v2 in X & v1<>v & v2<>v & v1<>v2
proof
 let X be set, v be object;
 assume 3 c= card X;
 then consider x,y,z being object such that
A1: x in X and
A2: y in X and
A3: z in X and
A4: x<>y and
A5: x<>z and
A6: y<>z by PENCIL_1:5;
   v <> x & v <> y & v <> z or v = x or v = y or v = z;
  hence ex v1, v2 being object
   st v1 in X & v2 in X & v1<>v & v2<>v & v1<>v2 by A1,A2,A3,A4,A5,A6;
end;
