
theorem Th108: :: MClique1
for G being with_finite_clique# SimpleGraph st clique# G = 1
for D being finite Clique of Mycielskian G holds order D <= 2
proof
  let G be with_finite_clique# SimpleGraph such that
A1: clique# G = 1;
   set uG = union G; set MG = Mycielskian G; set uMG = union MG;
  let D be finite Clique of Mycielskian G;
   set uD = union D;
  assume A2: order D > 2;
   then A3: order D >= 2+1 by NAT_1:13;
   uD is non empty by A2;
   then consider v being object such that
A4: v in uD;
A5: v <> uG by A4,A3,Th105;
   Segm 3 c= Segm order D by A3,NAT_1:39;
   then consider v1, v2 being object such that
A6: v1 in uD and v2 in uD and
A7: v1<>v and v2<>v and v1<>v2 by Th5;
A8: v1 <> uG by A6,A3,Th105;
    set e = {v,v1};
    e in D by A4,A6,Th53;
    then A9: e in Edges MG by A7,Th12;
    per cases by A9,Th90;
    suppose e in Edges G;
     hence contradiction by A1,Th57;
    end;
    suppose ex x, y being Element of union G
             st e = {x,[y,uG]} & {x,y} in Edges G;
       then consider x, y being Element of uG such that e = {x,[y,uG]} and
    A10: {x,y} in Edges G;
     thus contradiction by A1,A10,Th57;
    end;
    suppose ex y being Element of union G st e = {uG,[y,uG]} & y in uG;
      then consider y being Element of uG such that
    A11: e = {uG,[y,uG]} and y in uG;
     thus contradiction by A5,A8,A11,ZFMISC_1:6;
    end;
end;
