
theorem Th94:
for G being SimpleGraph, u being object
 st { union G, u } in Edges Mycielskian G
  ex x being object st x in union G & u = [x, union G]
proof
   let G be SimpleGraph, u be object such that
A1: { union G, u } in Edges Mycielskian G;
   set uG = union G;
   per cases by A1,Th93;
   suppose {uG, u} in Edges G;
     then uG in uG by Th13;
     hence ex x being object st x in uG & u = [x, uG];
   end;
   suppose (uG in uG or uG = uG)
           & (ex y being object st y in uG & u = [y,uG]);
     hence ex x being object st x in uG & u = [x, union G];
   end;
   suppose (u in uG or u = uG)
           & (ex y being object st y in uG & uG = [y,uG]);
       then consider y being set such that y in uG and
   A2: uG = [y,uG];
     thus ex x being object st x in uG & u = [x, uG] by A2,Th2;
   end;
end;
