Let x0 of type ι be given.
Apply unknownprop_f23dde3020cfe827bdc4db0338b279dd2c0f6c90742a195a1a7a614475669076 with
λ x1 . mul_nat x0 x1 = mul_nat x1 x0 leaving 2 subgoals.
Apply unknownprop_73039c0d651fbcf682cadad5d206a2e90632cd5b8241df76c9856acb60ebd224 with
x0,
λ x1 x2 . mul_nat x0 0 = x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying unknownprop_4756ca8c34efd0461ee4f316febaf4ee77ac8f03a3f9f75c481b60c5f8500b17 with x0.
Let x1 of type ι be given.
Apply unknownprop_3defe724d02ba276d9730f9f5a87e86e6bc0e48da350a99bdaaf13f339867dcf with
x0,
x1,
λ x2 x3 . x3 = mul_nat (ordsucc x1) x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply unknownprop_f3c14955b8925cb2239ff5fd3d7717614370df79ba39b9db8770a7884f7ded7e with
x1,
x0,
λ x2 x3 . add_nat x0 (mul_nat x0 x1) = x3 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Apply H2 with
λ x2 x3 . add_nat x0 x3 = add_nat (mul_nat x1 x0) x0.
Apply unknownprop_0bc05861c326df1bc856e9663c3bd091fe6729155095d255366b172bf07600be with
x0,
mul_nat x1 x0 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply unknownprop_0b229518762ed7010020950c24a2d0fe47c44c7a7b255cdddc862baf12395763 with
x1,
x0 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.