 reserve x for Real,
    p,k,l,m,n,s,h,i,j,k1,t,t1 for Nat,
    X for Subset of REAL;
reserve x for object, X,Y,Z for set;
 reserve M,N for Cardinal;
reserve X for non empty set,
  s for sequence of X;

theorem
  for X being finite set st 1 < card X holds ex x1,x2 being set st
  x1 in X & x2 in X & x1 <> x2
proof
  let X be finite set;
  set x1 = the Element of X;
  assume
A1: 1 < card X;
  then X <> {};
  then
A2: x1 in X;
  now
    assume
A3: for x2 being set st x2 in X holds x2 = x1;
    now
      let x be object;
      hereby
        assume x in X;
        then x = x1 by A3;
        hence x in {x1} by TARSKI:def 1;
      end;
      assume x in {x1};
      hence x in X by A2,TARSKI:def 1;
    end;
    then X = {x1} by TARSKI:2;
    hence contradiction by A1,CARD_1:30;
  end;
  hence thesis;
end;
