reserve X for set;

theorem Th26:
  for A being set st A is_SetOfSimpleGraphs_of X holds
    SIMPLEGRAPHS(X) c= A
proof
  let OTHER be set;
  defpred X[set] means $1 in OTHER;
  assume
A1: OTHER is_SetOfSimpleGraphs_of X;
A2: for g being SimpleGraph of X, v being set st g in SIMPLEGRAPHS(X) & X[g]
& v in X & not v in (the carrier of g) holds X[SimpleGraphStruct (#(the carrier
    of g)\/{v}, {}TWOELEMENTSETS((the carrier of g)\/{v})#)]
  proof
    let g be SimpleGraph of X, v be set;
    assume that
    g in SIMPLEGRAPHS(X) and
A3: g in OTHER & v in X & not v in (the carrier of g);
    set SVg=(the carrier of g);
    SVg is Subset of X by Th21;
    hence thesis by A1,A3;
  end;
A4: for g being SimpleGraph of X, e being set st X[g] & e in TWOELEMENTSETS(
  the carrier of g) & not e in (the SEdges of g) holds ex sege being Subset of
  TWOELEMENTSETS(the carrier of g) st sege=((the SEdges of g)\/{e}) & X[
  SimpleGraphStruct (#(the carrier of g),sege#)]
  proof
    let g be SimpleGraph of X, e be set;
    assume that
A5: g in OTHER and
A6: e in TWOELEMENTSETS(the carrier of g) and
A7: not e in (the SEdges of g);
    set SVg = (the carrier of g), SEg = (the SEdges of g);
    consider v1,v2 being object such that
A8: v1 in SVg & v2 in SVg & v1<>v2 and
A9: e={v1,v2} by A6,Th8;
    SVg is finite Subset of X by Th21;
    then consider v1v2 being finite Subset of TWOELEMENTSETS(SVg) such that
A10: v1v2=(SEg \/ {{v1,v2}}) & SimpleGraphStruct (#SVg,v1v2#) in OTHER
    by A1,A5,A7,A8,A9;
    take v1v2;
    thus thesis by A9,A10;
  end;
  let e be object;
  assume
A11: e in SIMPLEGRAPHS(X);
A12: X[SimpleGraphStruct (#{},{}TWOELEMENTSETS{}#)] by A1;
  for e being set st e in SIMPLEGRAPHS(X) holds X[e] from
  IndSimpleGraphs0(A12,A2,A4);
  hence thesis by A11;
end;
