summation_add: (all a:Nat. (all b:Nat, s:Nat, t:Nat, f:(fn Nat -> Nat), g:(fn Nat -> Nat), h:(fn Nat -> Nat). (if ((all i:Nat. (if i < a then g(s + i) = f(s + i))) and (all i:Nat. (if i < b then h(t + i) = f((s + a) + i)))) then summation(a + b, s, f) = summation(a, s, g) + summation(b, t, h))))

summation_cong: (all k:Nat. (all f:(fn Nat -> Nat), g:(fn Nat -> Nat), s:Nat, t:Nat. (if (all i:Nat. (if i < k then f(s + i) = g(t + i))) then summation(k, s, f) = summation(k, t, g))))