
theorem Th86:
for G being SimpleGraph
 holds Vertices Mycielskian G = union G \/ [:union G,{union G}:] \/ {union G}
proof
 let G be SimpleGraph;
 set uG = union G; set MG = Mycielskian G; set uMG = union MG;
A1: uG in {uG} by TARSKI:def 1;
 thus uMG c= uG \/ [:uG,{uG}:] \/ {uG} proof
    let v be object;
    assume A2: v in uMG;
    per cases by A2,Th85;
    suppose v in uG;
      then v in uG \/ ([:uG,{uG}:] \/ {uG}) by XBOOLE_0:def 3;
     hence v in uG \/ [:uG,{uG}:] \/ {uG} by XBOOLE_1:4;
    end;
    suppose ex x being set st x in uG & v = [x,uG];
      then consider x being set such that
    A3: x in uG and
    A4: v = [x,uG];
      v in [:uG,{uG}:] by A1,A3,A4,ZFMISC_1:def 2;
      then v in uG \/ [:uG,{uG}:] by XBOOLE_0:def 3;
      hence v in uG \/ [:uG,{uG}:] \/ {uG} by XBOOLE_0:def 3;
    end;
    suppose v = union G; then
     v in {uG} by TARSKI:def 1;
     hence thesis by XBOOLE_0:def 3;
    end;
 end;
 thus uG \/ [:uG,{uG}:] \/ {uG} c= uMG proof
  let v be object;
  assume v in uG \/ [:uG,{uG}:] \/ {uG};
  then A5: v in uG \/ [:uG,{uG}:] or v in {uG} by XBOOLE_0:def 3;
  per cases by A5,XBOOLE_0:def 3;
  suppose v in uG;
   hence thesis by Th85;
  end;
  suppose v in [:uG,{uG}:];
    then consider x, y being object such that
    A6: x in uG and
    A7: y in {uG} and
    A8: v = [x,y] by ZFMISC_1:def 2;
       y = uG by A7,TARSKI:def 1;
   hence thesis by A6,A8,Th85;
  end;
  suppose v in {uG};
    then v = uG by TARSKI:def 1;
   hence thesis by Th85;
  end;
 end;
end;
