
theorem Th91:
for G being SimpleGraph
 holds Edges Mycielskian G = (Edges G)
  \/ { {x,[y,union G]} where x, y is Element of union G : {x,y} in Edges G }
  \/ { {union G,[y,union G]} where y is Element of union G : y in union G }
proof
 let G be SimpleGraph;
 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 };
 thus Edges Mycielskian G c= (Edges G) \/ A \/ B
 proof let e be object; assume
 A1: e in Edges Mycielskian G;
  per cases by A1,Th90;
  suppose e in Edges G;
     then e in (Edges G) \/ A by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose ex x, y being Element of union G
           st e = {x,[y,union G]} & {x,y} in Edges G;
     then e in A;
     then e in (Edges G) \/ A by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose ex y being Element of union G st e = {uG,[y,uG]} & y in uG;
     then e in B;
   hence thesis by XBOOLE_0:def 3;
  end;
 end;

 thus (Edges G) \/ A \/ B c= Edges Mycielskian G proof
  let e be object;
  assume e in (Edges G) \/ A \/ B;
  then A2: e in (Edges G) \/ A or e in B by XBOOLE_0:def 3;
  per cases by A2,XBOOLE_0:def 3;
  suppose e in Edges G;
   hence thesis by Th90;
  end;
  suppose e in A;
    then consider x, y being Element of uG such that
  A3: e = {x,[y,uG]} & {x,y} in Edges G;
   thus thesis by A3,Th90;
  end;
  suppose e in B;
   then consider y being Element of uG such that
  A4: e = {union G,[y,union G]} & y in uG;
   thus thesis by A4,Th90;
  end;
 end;
end;
