
theorem Th12:
  for f,g being Function, a,A being set st rng f c= dom f & a in dom f holds
  a nin A implies ((A,g) iter f).a = a
proof
  let f,g be Function;
  let a,A be set;
  assume that
A1: rng f c= dom f and
A2: a in dom f and
A3: a nin A;
  field f = dom f by A1,XBOOLE_1:12;
  then iter(f,0) = id dom f by FUNCT_7:68;
  then
A4: a = iter(f,0).a by A2,FUNCT_1:18;
  for i being Nat st i < 0 holds iter(f,i).a in A;
  hence thesis by A1,A2,A3,A4,Def7;
end;
