
theorem Th88:
for G being void SimpleGraph holds Mycielskian G = {{},{union G}}
proof
  let G be void SimpleGraph;
  set uG = union G;
A1: { {uG,[x,uG]} where x is Element of uG : x in Vertices G } = {} proof
     assume not thesis;
       then consider e being object such that
   A2: e in { {uG,[x,uG]} where x is Element of uG : x in Vertices G }
       by XBOOLE_0:def 1;
       consider x being Element of uG such that e = {uG,[x,uG]} and
   A3: x in Vertices G by A2;
     thus thesis by A3;
   end;
A4: { {x,[y,uG]} where x, y is Element of uG : {x,y} in Edges G } = {} proof
     assume not thesis;
      then consider e being object such that
     A5: e in { {x,[y,uG]} where x, y is Element of uG : {x,y} in Edges G }
                     by XBOOLE_0:def 1;
         consider x, y being Element of uG such that e = {x,[y,uG]} and
     A6: {x,y} in Edges G by A5;
     thus thesis by A6,Th13;
   end;
A7: Edges G = {} proof
    assume not thesis;
    then consider e being object such that
   A8: e in Edges G by XBOOLE_0:def 1;
       consider x, y being set such that x <> y and
   A9: x in Vertices G and y in Vertices G & e = {x, y} by A8,Th11;
    thus contradiction by A9;
   end;
A10: the set of all {x} where x is Element of (uG) \/ [:uG,{uG}:] \/ {uG}
              = {{uG}} proof
    thus the set of all {x} where x is Element of (uG) \/ [:uG,{uG}:] \/ {uG}
           c= {{uG}} proof
      let a be object;
      assume a in the set of all
 {x} where x is Element of (uG) \/ [:uG,{uG}:] \/ {uG};
        then consider x being Element of (uG) \/ [:uG,{uG}:] \/ {uG} such that
      A11: a = {x};
        x = uG by TARSKI:def 1;
     hence a in {{uG}} by A11,TARSKI:def 1;
    end;
    thus {{uG}} c= the set of all
 {x} where x is Element of (uG) \/ [:uG,{uG}:] \/ {uG}
              proof
      let a be object;
      assume a in {{uG}};
        then A12: a = {uG} by TARSKI:def 1;
        uG in (uG) \/ [:uG,{uG}:] \/ {uG} by TARSKI:def 1;
      hence thesis by A12;
    end;
   end;
 thus Mycielskian G = {{},{uG}} by A1,A4,A7,A10,ENUMSET1:1;
end;
