reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve R for Ring;
reserve G,H for LeftMod of R;
reserve V for LeftMod_DOMAIN of R;

theorem
  for UN
   ex x being object st x in the set of all
[G,f] where G is Element of GroupObjects(UN)
, f is Element of Funcs([:the carrier of R,{{}}:],{{}})
   & GO x, TrivialLMod(R),R
proof
  let UN;
  set T = TrivialLMod(R);
  reconsider x1 = the addLoopStr of T as Element of GroupObjects(UN)
      by GRCAT_1:29;
  reconsider f1 = the lmult of T as Function of [:the carrier of R,{{}}:],{{}};
  reconsider x2 = f1 as Element of Funcs([:the carrier of R,{{}}:],{{}})
        by FUNCT_2:8;
  take x = [x1,x2];
  thus x in the set of all
[G,f] where G is Element of GroupObjects(UN), f is Element of
  Funcs([:the carrier of R,{{}}:],{{}}) ;
  thus thesis;
end;
