
theorem Th99:
for G being SimpleGraph, x, y being object
 st {[x,union G], y} in Edges Mycielskian G
  holds x <> y & x in union G &  (y in union G or y = union G)
proof
  let G be SimpleGraph, x, y be object such that
A1: {[x,union G], y} in Edges Mycielskian G;
   set uG = union G; set e = {[x,union G],y};
   per cases by A1,Th90;
   suppose e in Edges G;
      then [x,uG] in uG by Th13;
    hence x<>y & x in union G &  (y in union G or y = union G) by Th1;
  end;
  suppose ex x, y being Element of uG
           st e = {x,[y,union G]} & {x,y} in Edges G;
    then consider xa, ya being Element of uG such that
  A2: e = {xa,[ya,union G]} and
  A3: {xa,ya} in Edges G;
     consider xx, yy being set such that
  A4: xx <> yy and
  A5: xx in Vertices G & yy in Vertices G and
  A6: {xa,ya} = {xx, yy} by A3,Th11;
  A7: xa = xx & ya = yy or xa =yy & ya = xx by A6,ZFMISC_1:6;
      per cases by A2,ZFMISC_1:6;
      suppose xa = [x,uG] & y = [ya,uG];
       hence thesis by A5,Th1;
      end;
      suppose xa = y & [ya,uG] = [x,uG];
    hence x<>y & x in union G & (y in union G or y = union G)
     by A4,A5,A7,XTUPLE_0:1;
      end;
  end;
  suppose ex y being Element of union G
           st e = {union G,[y,union G]} & y in union G;
     then consider yy being Element of uG such that
  A8: e = {union G,[yy,union G]} and
  A9: yy in union G;
  A10: uG = [x,uG] & y = [yy,uG] or uG = y & [x,uG] = [yy, uG]
       by A8,ZFMISC_1:6;
      x = yy by A10,Th2,XTUPLE_0:1;
    hence x<>y & x in union G & (y in union G or y = union G) by A10,A9;
  end;
end;
