
theorem Th3:
  for X being set st 2 c= card X for x being object ex y being object st
  y in X & x<>y
proof
  let X be set;
  assume 2 c= card X;
  then consider a,b being object such that
A1: a in X and
A2: b in X & a <> b by Th2;
  let x be object;
  per cases;
  suppose
A3: x=a;
    take b;
    thus thesis by A2,A3;
  end;
  suppose
A4: x <> a;
    take a;
    thus thesis by A1,A4;
  end;
end;
