
theorem Th105:
for G being SimpleGraph, C being finite Clique of Mycielskian G
 st 3 <= order C for v being Vertex of C holds v <> union G
proof
  let G be SimpleGraph, C be finite Clique of Mycielskian G such that
A1: 3 <= order C;
  set MG = Mycielskian G;
  let v be Vertex of C such that
A2: v = union G;
   Segm 3 c= Segm order C by A1,NAT_1:39;
   then consider v1, v2 being object such that
A3: v1 in Vertices C and
A4: v2 in Vertices C and
A5: v1<>v and
A6: v2<>v and
A7: v1<>v2 by Th5;
A8: {v,v1} in C by A3,Th53;
A9: {v,v2} in C by A4,Th53;
A10: {v, v1} in Edges MG by A8,A5,Th12;
A11: {v, v2} in Edges MG by A6,A9,Th12;
    consider x1 being object such that x1 in union G and
A12: v1 = [x1, union G] by A2,A10,Th94;
    consider x2 being object such that x2 in union G and
A13: v2 = [x2, union G] by A2,A11,Th94;
    {v1, v2} in C by A3,A4,Th53;
  hence contradiction by A12,A13,A7,Th98;
end;
