reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  (for f,g st dom f = X & dom g = X holds f = g) implies X = {}
proof
  deffunc F(object) = {};
  assume
A1: for f,g st dom f = X & dom g = X holds f = g;
  set x = the Element of X;
  consider f being Function such that
A2: dom f = X and
A3: for x st x in X holds f.x = F(x) from Lambda;
  assume
A4: not thesis;
  then
A5: f.x = {} by A3;
  deffunc F(object) = {{}};
  consider g being Function such that
A6: dom g = X and
A7: for x st x in X holds g.x = F(x) from Lambda;
  g.x = {{}} by A4,A7;
  hence contradiction by A1,A2,A6,A5;
end;
