reserve A,A1,A2,B,C,D for Ordinal,
  X,Y for set,
  x,y,a,b,c for object,
  L,L1,L2,L3 for Sequence,
  f for Function;

theorem
  inf X c= sup X
proof
  let x be object;
  set y = the Element of On X;
  assume
A1: x in inf X;
  then reconsider y as Ordinal by ORDINAL1:def 9,SETFAM_1:1;
  On X c= sup X by Def3;
  then y in sup X by A1,SETFAM_1:1;
  then
A2: y c= sup X by ORDINAL1:def 2;
  x in y by A1,SETFAM_1:1,def 1;
  hence thesis by A2;
end;
