Let x0 of type ι be given.
Let x1 of type ι be given.
Apply unknownprop_f23dde3020cfe827bdc4db0338b279dd2c0f6c90742a195a1a7a614475669076 with
λ x2 . mul_nat (mul_nat x0 x1) x2 = mul_nat x0 (mul_nat x1 x2) leaving 2 subgoals.
Apply unknownprop_4756ca8c34efd0461ee4f316febaf4ee77ac8f03a3f9f75c481b60c5f8500b17 with
x1,
λ x2 x3 . mul_nat (mul_nat x0 x1) 0 = mul_nat x0 x3.
Apply unknownprop_4756ca8c34efd0461ee4f316febaf4ee77ac8f03a3f9f75c481b60c5f8500b17 with
x0,
λ x2 x3 . mul_nat (mul_nat x0 x1) 0 = x3.
The subproof is completed by applying unknownprop_4756ca8c34efd0461ee4f316febaf4ee77ac8f03a3f9f75c481b60c5f8500b17 with
mul_nat x0 x1.
Let x2 of type ι be given.
Apply unknownprop_3defe724d02ba276d9730f9f5a87e86e6bc0e48da350a99bdaaf13f339867dcf with
mul_nat x0 x1,
x2,
λ x3 x4 . x4 = mul_nat x0 (mul_nat x1 (ordsucc x2)) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_3defe724d02ba276d9730f9f5a87e86e6bc0e48da350a99bdaaf13f339867dcf with
x1,
x2,
λ x3 x4 . add_nat (mul_nat x0 x1) (mul_nat (mul_nat x0 x1) x2) = mul_nat x0 x4 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_7f023976a8daf041aa17d3220ad6430692e696ae1148f2cbd9b38476d90b1552 with
x0,
x1,
mul_nat x1 x2,
λ x3 x4 . add_nat (mul_nat x0 x1) (mul_nat (mul_nat x0 x1) x2) = x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply unknownprop_0b229518762ed7010020950c24a2d0fe47c44c7a7b255cdddc862baf12395763 with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.