
theorem
  for X being non empty set, x being Element of X ex f being Enumeration
  of X st f.x = 0
proof
  let X be non empty set, x be Element of X;
  set f = the Enumeration of X;
A1: 0 in card X by ORDINAL3:8;
A2: rng f = card X by Th6;
  dom f = X by Th6;
  then consider y being object such that
A3: y in X and
A4: 0 = f.y by A1,A2,FUNCT_1:def 3;
  reconsider y as Element of X by A3;
  reconsider g = f+*(y,f.x)+*(x,0) as Enumeration of X by A4,Th8;
  take g;
  dom f = X by Th6;
  then dom (f+*(y,f.x)) = X by FUNCT_7:30;
  hence thesis by FUNCT_7:31;
end;
