Let x0 of type ι be given.
Let x1 of type ι be given.
Apply unknownprop_f23dde3020cfe827bdc4db0338b279dd2c0f6c90742a195a1a7a614475669076 with
λ x2 . mul_nat x0 (add_nat x1 x2) = add_nat (mul_nat x0 x1) (mul_nat x0 x2) leaving 2 subgoals.
Apply unknownprop_bad5adbbba30ab6e9c584ed350d824b3c3bff74e61c0a5380ac75f32855c37ee with
x1,
λ x2 x3 . mul_nat x0 x3 = add_nat (mul_nat x0 x1) (mul_nat x0 0).
Apply unknownprop_4756ca8c34efd0461ee4f316febaf4ee77ac8f03a3f9f75c481b60c5f8500b17 with
x0,
λ x2 x3 . mul_nat x0 x1 = add_nat (mul_nat x0 x1) x3.
Let x2 of type ι → ι → ο be given.
The subproof is completed by applying unknownprop_bad5adbbba30ab6e9c584ed350d824b3c3bff74e61c0a5380ac75f32855c37ee with
mul_nat x0 x1,
λ x3 x4 . x2 x4 x3.
Let x2 of type ι be given.
Apply unknownprop_bfc870f6d786cc78805c5bf0f9864161d18f532f6daf7daf1d02f4a58dac06f9 with
x1,
x2,
λ x3 x4 . mul_nat x0 x4 = add_nat (mul_nat x0 x1) (mul_nat x0 (ordsucc x2)) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_3defe724d02ba276d9730f9f5a87e86e6bc0e48da350a99bdaaf13f339867dcf with
x0,
add_nat x1 x2,
λ x3 x4 . x4 = add_nat (mul_nat x0 x1) (mul_nat x0 (ordsucc x2)) leaving 2 subgoals.
Apply unknownprop_3336121954edce0fefb5edee2ad1b426a9827aac09625122db0ff807b493dc73 with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply H3 with
λ x3 x4 . add_nat x0 x4 = add_nat (mul_nat x0 x1) (mul_nat x0 (ordsucc x2)).
Apply unknownprop_3defe724d02ba276d9730f9f5a87e86e6bc0e48da350a99bdaaf13f339867dcf with
x0,
x2,
λ x3 x4 . add_nat x0 (add_nat (mul_nat x0 x1) (mul_nat x0 x2)) = add_nat (mul_nat x0 x1) x4 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_49efa1ef20aa85e1a1857df356e1a1da17f23d968b35d2278ca5633fe8adcf44 with
x0,
mul_nat x0 x1,
mul_nat x0 x2,
λ x3 x4 . x3 = add_nat (mul_nat x0 x1) (add_nat x0 (mul_nat x0 x2)) leaving 3 subgoals.
Apply unknownprop_0b229518762ed7010020950c24a2d0fe47c44c7a7b255cdddc862baf12395763 with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply unknownprop_0b229518762ed7010020950c24a2d0fe47c44c7a7b255cdddc862baf12395763 with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply unknownprop_0bc05861c326df1bc856e9663c3bd091fe6729155095d255366b172bf07600be with
x0,
mul_nat x0 x1,
λ x3 x4 . add_nat x4 (mul_nat x0 x2) = add_nat (mul_nat x0 x1) (add_nat x0 (mul_nat x0 x2)) leaving 3 subgoals.
The subproof is completed by applying H0.
Apply unknownprop_0b229518762ed7010020950c24a2d0fe47c44c7a7b255cdddc862baf12395763 with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply unknownprop_49efa1ef20aa85e1a1857df356e1a1da17f23d968b35d2278ca5633fe8adcf44 with
mul_nat x0 x1,
x0,
mul_nat x0 x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply unknownprop_0b229518762ed7010020950c24a2d0fe47c44c7a7b255cdddc862baf12395763 with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.