reserve X for set;

theorem Th24:
  for V being Subset of X, E being Subset of TWOELEMENTSETS(V),
  v1,v2 being set st v1 in V & v2 in V & v1<>v2 &
  SimpleGraphStruct (#V,E#) in SIMPLEGRAPHS(X) holds
  ex v1v2 being finite Subset of TWOELEMENTSETS(V) st
  v1v2 = (E \/ {{v1,v2}}) & SimpleGraphStruct (#V,v1v2#) in SIMPLEGRAPHS(X)
proof
  let V be Subset of X, E be Subset of TWOELEMENTSETS(V), v1,v2 be set;
  set g = SimpleGraphStruct (#V,E#);
  assume that
A1: v1 in V & v2 in V and
A2: not v1=v2 and
A3: g in SIMPLEGRAPHS(X);
  reconsider g as SimpleGraph of X by A3,Def4;
A4: (the SEdges of g) is finite Subset of TWOELEMENTSETS(V) by Th21;
  (the carrier of g) is finite Subset of X by Th21;
  then reconsider V as finite Subset of X;
  (E \/ {{v1,v2}}) c= TWOELEMENTSETS(V) & (E \/ {{v1,v2}}) is finite
  proof
    hereby
      let e be object;
      assume
A5:   e in E \/ {{v1,v2}};
      per cases by A5,XBOOLE_0:def 3;
      suppose
        e in E;
        hence e in TWOELEMENTSETS(V);
      end;
      suppose
        e in {{v1,v2}};
        then
A6:     e={v1,v2} by TARSKI:def 1;
        then e is Subset of V by A1,ZFMISC_1:32;
        hence e in TWOELEMENTSETS(V) by A1,A2,A6,Th8;
      end;
    end;
    thus thesis by A4;
  end;
  then reconsider E9 = (E \/ {{v1,v2}}) as finite Subset of TWOELEMENTSETS(V);
  SimpleGraphStruct (#V,E9#) in SIMPLEGRAPHS(X);
  hence thesis;
end;
