
theorem Th6:
  for X being set st 3 c= card X for x,y being object ex z being object
  st z in X & x<>z & y<>z
proof
  let X be set;
  assume 3 c= card X;
  then consider a,b,c being object such that
A1: a in X and
A2: b in X and
A3: c in X and
A4: a<>b and
A5: a<>c & b<>c by Th5;
  let x,y be object;
  per cases;
  suppose
    x <> a & y <> a;
    hence thesis by A1;
  end;
  suppose
    x <> a & y = a & x = b;
    hence thesis by A3,A5;
  end;
  suppose
    x <> a & y = a & x <> b;
    hence thesis by A2,A4;
  end;
  suppose
    x = a & y <> a & y=b;
    hence thesis by A3,A5;
  end;
  suppose
    x = a & y <> a & y<>b;
    hence thesis by A2,A4;
  end;
  suppose
    x = a & y = a;
    hence thesis by A2,A4;
  end;
end;
