
theorem Th90:
for G being SimpleGraph, e being set
 holds e in Edges Mycielskian G iff
  e in Edges G or
  (ex x, y being Element of union G st e = {x,[y,union G]} & {x,y} in Edges G)
or (ex y being Element of union G st e = {union G,[y,union G]} & y in union G)
proof
 let G be SimpleGraph, e be set;
 set uG = union G; set MG = Mycielskian G;
 set C = the set of all {x} where x is Element of (uG) \/ [:uG,{uG}:] \/ {uG};
 set A = { {x,[y,uG]} where x, y is Element of uG : {x,y} in Edges G };
 set B = { {uG,[x,uG]} where x is Element of uG : x in Vertices G };
 hereby assume A1: e in Edges Mycielskian G;
  then consider x, y being set such that
 A2: x <> y and x in Vertices MG and y in Vertices MG and
 A3: e = {x, y} by Th11;
  per cases by A1,MYCIELSK:4;
  suppose e in { {} };
   hence e in Edges G
    or (ex x, y being Element of uG st e = {x,[y,uG]} & {x,y} in Edges G)
 or (ex y being Element of uG st e = {uG,[y,uG]} & y in uG)
   by A3,TARSKI:def 1;
  end;
  suppose e in C;
    then consider a being Element of (uG) \/ [:uG,{uG}:] \/ {uG} such that
    A4: e = {a};
   thus e in Edges G
    or (ex x, y being Element of uG st e = {x,[y,uG]} & {x,y} in Edges G)
    or (ex y being Element of uG st e = {uG,[y,uG]} & y in uG)
        by A4,A3,A2,ZFMISC_1:5;
  end;
  suppose e in Edges G or e in A or e in B;
   hence e in Edges G
    or (ex x, y being Element of uG st e = {x,[y,uG]} & {x,y} in Edges G)
   or (ex y being Element of uG st e = {uG,[y,uG]} & y in uG);
  end;
 end;
 assume A5: e in Edges G
    or (ex x, y being Element of uG st e = {x,[y,uG]} & {x,y} in Edges G)
    or (ex y being Element of uG st e = {uG,[y,uG]} & y in uG);
  per cases by A5;
  suppose A6: e in Edges G;
  A7: card e = 2 by A6,Def1;
      e in MG by A6,MYCIELSK:4;
   hence e in Edges Mycielskian G by A7,Def1;
  end;
  suppose ex x, y being Element of uG st e = {x,[y,uG]} & {x,y} in Edges G;
    then consider x, y being Element of uG such that
  A8: e = {x,[y,uG]} and
  A9: {x,y} in Edges G;
  A10: e in A by A8,A9;
  A11: e in MG by A10,MYCIELSK:4;
      y in uG by A9,Th13;
      then x <> [y,uG] by Th1;
      then card e = 2 by A8,CARD_2:57;
   hence e in Edges Mycielskian G by A11,Def1;
  end;
  suppose ex y being Element of uG st e = {uG,[y,uG]} & y in uG;
    then consider y being Element of uG such that
  A12: e = {uG,[y,uG]} and
  A13: y in uG;
  A14: e in B by A12,A13;
  A15: e in MG by A14,MYCIELSK:4;
      card e = 2 by Th2,A12,CARD_2:57;
   hence e in Edges Mycielskian G by A15,Def1;
  end;
end;
