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Proofgold Proof
pf
Assume H0:
prime_nat
4
.
Apply H0 with
False
.
Assume H1:
and
(
4
∈
omega
)
(
1
∈
4
)
.
Assume H2:
∀ x0 .
x0
∈
omega
⟶
divides_nat
x0
4
⟶
or
(
x0
=
1
)
(
x0
=
4
)
.
Apply H2 with
2
,
False
leaving 4 subgoals.
Apply nat_p_omega with
2
.
The subproof is completed by applying nat_2.
Apply and3I with
2
∈
omega
,
4
∈
omega
,
∃ x0 .
and
(
x0
∈
omega
)
(
mul_nat
2
x0
=
4
)
leaving 3 subgoals.
Apply nat_p_omega with
2
.
The subproof is completed by applying nat_2.
Apply nat_p_omega with
4
.
The subproof is completed by applying nat_4.
Let x0 of type
ο
be given.
Assume H3:
∀ x1 .
and
(
x1
∈
omega
)
(
mul_nat
2
x1
=
4
)
⟶
x0
.
Apply H3 with
2
.
Apply andI with
2
∈
omega
,
mul_nat
2
2
=
4
leaving 2 subgoals.
Apply nat_p_omega with
2
.
The subproof is completed by applying nat_2.
The subproof is completed by applying unknownprop_74ac8a784913fa4a6f9da3de96c05984e11ff1600ef66d703e49d6ee22ad106d.
The subproof is completed by applying neq_2_1.
Assume H3:
2
=
4
.
Apply neq_4_2.
Let x0 of type
ι
→
ι
→
ο
be given.
The subproof is completed by applying H3 with
λ x1 x2 .
x0
x2
x1
.
■