have h10 : c / (a + b + c + d) < c / (b + c + d) :=
   div_lt_div_of_pos_left hc (by linarith) (by linarith)
case intro.intro.intro.intro.intro.intro.intro.intro.left
a b c d : ℝ
ha : 0 < a
hb : 0 < b
hc : 0 < c
hd : 0 < d
h8 : a / (a + b + c + d) < a / (a + b + d)
h9 : b / (a + b + c + d) < b / (a + b + c)
⊢ 1 < a / (a + b + d) + b / (a + b + c) + c / (b + c + d) + d / (a + c + d)
case intro.intro.intro.intro.intro.intro.intro.intro.left
a b c d : ℝ
ha : 0 < a
hb : 0 < b
hc : 0 < c
hd : 0 < d
h8 : a / (a + b + c + d) < a / (a + b + d)
h9 : b / (a + b + c + d) < b / (a + b + c)
h10 : c / (a + b + c + d) < c / (b + c + d)
⊢ 1 < a / (a + b + d) + b / (a + b + c) + c / (b + c + d) + d / (a + c + d)