reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;
reserve f for Function of A,B;

theorem
  card B in card A & B <> {} implies ex x,y st x in A & y in A & x <> y
  & f.x = f.y
proof
  assume that
A1: card B in card A and
A2: B <> {} and
A3: for x,y st x in A & y in A & x <> y holds f.x <> f.y;
A4: dom f = A by A2,FUNCT_2:def 1;
  then for x,y being object holds x in dom f & y in dom f & f.x = f.y
implies x = y by A3;
  then f is one-to-one;
  then dom f,f .: (dom f) are_equipotent by CARD_1:33;
  then dom f,rng f are_equipotent by RELAT_1:113;
  then
A5: card A = card(rng f) by A4,CARD_1:5;
  rng f c= B by RELAT_1:def 19;
  then card A c= card B by A5,CARD_1:11;
  hence contradiction by A1,CARD_1:4;
end;
