reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem Th35:
  for f be Function of Segm(n+1),Segm k st k<>0 & f"{f.n}<>{n}
   ex m st m in f"{f.n} & m<>n
proof
  let f be Function of Segm(n+1), Segm k such that
A1: k<>0 and
A2: f"{f.n}<>{n};
A3: n<n+1 by NAT_1:13;
A4: f.n in {f.n} by TARSKI:def 1;
  dom f =n+1 by A1,FUNCT_2:def 1;
  then n in dom f by A3,NAT_1:44;
  then n in f"{f.n} by A4,FUNCT_1:def 7;
  then ex m be object st m in f"{f.n} & m<>n by A2,ZFMISC_1:35;
  hence thesis;
end;
