reserve X for set;

theorem Th23:
  for V being Subset of X, E being Subset of TWOELEMENTSETS(V),
  n being set, Evn being finite Subset of TWOELEMENTSETS(V \/ {n}) st
  SimpleGraphStruct (#V,E#) in SIMPLEGRAPHS(X) & n in X holds
  SimpleGraphStruct (#(V \/ {n}),Evn#) in SIMPLEGRAPHS(X)
proof
  let V be Subset of X, E be Subset of TWOELEMENTSETS(V), n be set, Evn be
  finite Subset of TWOELEMENTSETS(V \/ {n});
  set g = SimpleGraphStruct (#V,E#);
  assume that
A1: g in SIMPLEGRAPHS(X) and
A2: n in X;
  reconsider g as SimpleGraph of X by A1,Def4;
  V = (the carrier of g);
  then V is finite Subset of X by Th21;
  then (V \/ {n}) is finite Subset of X by A2,Lm1;
  hence thesis;
end;
