
theorem Th101:
for G being SimpleGraph, x, y being set
 st y in union G & {[x,union G], y} in Mycielskian G holds {x, y} in G
proof
  let G be SimpleGraph;
  set MG = Mycielskian G; set uG = union G;
  set A = { {x,[y,uG]} where x, y is Element of uG : {x,y} in Edges G };
  set B = { {uG,[y,uG]} where y is Element of uG : y in uG };
  let x, y be set;
  assume A1: y in uG;
  assume {[x,uG], y} in MG;
  then {[x,uG], y} in { {} } \/ singletons Vertices MG \/ Edges MG by Th27;
  then A2: {[x,uG], y} in { {} } \/ singletons Vertices MG
           or {[x,uG], y} in Edges MG by XBOOLE_0:def 3;
  per cases by A2,XBOOLE_0:def 3;
  suppose {[x,uG], y} in { {} };
   hence thesis by TARSKI:def 1;
  end;
  suppose {[x,uG], y} in singletons Vertices MG;
     then consider f being Subset of Vertices MG such that
  A3: f = {[x,uG], y} and
  A4: f is 1-element;
     consider s being set such that s in Vertices MG and
  A5: f = {s} by A4,Th9;
  A6: card f = 1 by A5,CARD_1:30;
      y = [x,uG] by A6,A3,CARD_2:57;
   hence thesis by A1,Th1;
  end;
  suppose {[x,uG], y} in Edges MG;
   then {[x,uG], y} in (Edges G) \/ A \/ B by Th91;
   then A7: {[x,uG], y} in (Edges G) \/ A or {[x,uG], y} in B
        by XBOOLE_0:def 3;
   per cases by A7,XBOOLE_0:def 3;
   suppose {[x,uG], y} in Edges G;
      then [x,uG] in uG by Th13;
    hence thesis by Th1;
   end;
   suppose {[x,uG], y} in A;
     then consider xx, yy being Element of uG such that
   A8: {[x,uG], y} = {xx,[yy,uG]} and
   A9: {xx,yy} in Edges G;
   A10: xx in uG & yy in uG by A9,Th13;
       [x,uG] = xx & y = [yy,uG] or [x,uG] = [yy,uG] & y = xx
        by A8,ZFMISC_1:6;
    then x = yy & y = xx by A10,Th1,XTUPLE_0:1;
    hence thesis by A9;
   end;
   suppose {[x,uG], y} in B;
     then consider yy being Element of uG such that
   A11: {[x,uG], y} = {uG,[yy,uG]} and yy in uG;
       [x,uG] = uG & y = [yy,uG] or [x,uG] = [yy,uG] & y = uG
        by A11,ZFMISC_1:6;
    hence thesis by Th1,A1;
   end;
  end;
end;
