
theorem Th93:
for G being SimpleGraph, s, t being object
 st {s, t} in Edges Mycielskian G
  holds {s, t} in Edges G or
        (s in union G or s = union G)
           & (ex y being object st y in union G & t = [y,union G]) or
        (t in union G or t = union G)
           & (ex y being object st y in union G & s = [y,union G])
proof
  let G be SimpleGraph, s, t be object such that
A1: {s, t} in Edges Mycielskian G;
  set uG = union G;
   per cases by A1,Th90;
   suppose {s, t} in Edges G;
     hence thesis;
   end;
   suppose ex x, y being Element of uG
            st {s,t} = {x,[y,uG]} & {x,y} in Edges G;
      then consider x, y being Element of uG such that
   A2: {s,t} = {x,[y,uG]} and
   A3: {x,y} in Edges G;
   A4: x in uG & y in uG by A3,Th13;
       s = x & t = [y,uG] or t = x & s = [y,uG] by A2,ZFMISC_1:6;
     hence thesis by A4;
   end;
   suppose ex y being Element of uG st {s,t} = {uG,[y,uG]} & y in uG;
      then consider y being Element of uG such that
   A5: {s,t} = {uG,[y,uG]} and
   A6: y in uG;
       s = uG & t = [y,uG] or t = uG & s = [y,uG] by A5,ZFMISC_1:6;
     hence thesis by A6;
   end;
end;
