
theorem Th103:
for G being SimpleGraph, x, y being set
 st x in Vertices G & y in Vertices G & {x, y} in Mycielskian G
  holds {x, y} in G
proof
 let G be SimpleGraph, s, t be set such that
A1: s in Vertices G and
A2: t in Vertices G and
A3: {s, t} in Mycielskian G;
 per cases;
 suppose s = t;
   then {s,t} = {s} by ENUMSET1:29;
  hence {s, t} in G by A1,Th24;
 end;
 suppose s<>t;
  then card {s,t} = 2 by CARD_2:57;
  then A4: {s, t} in Edges Mycielskian G by A3,Def1;
  per cases by A4,Th93;
  suppose {s, t} in Edges G;
    hence {s, t} in G;
  end;
  suppose (s in union G or s = union G)
    & (ex y being object st y in union G & t = [y,union G]);
    then consider y be set such that y in union G and
  A5: t = [y,union G];
    thus {s, t} in G by A5,A2,Th1;
  end;
  suppose (t in union G or t = union G)
    & (ex y being object st y in union G & s = [y,union G]);
    then consider y being set such that y in union G and
  A6: s = [y,union G];
    thus {s, t} in G by A6,A1,Th1;
  end;
 end;
end;
