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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Let x2 of type
ι
be given.
Apply Repl_Empty with
λ x3 .
SetAdjoin
x3
(
Sing
4
)
,
λ x3 x4 .
binunion
(
binunion
(
binunion
x0
{
(
λ x6 .
SetAdjoin
x6
(
Sing
2
)
)
x5
|x5 ∈
x1
}
)
{
(
λ x6 .
SetAdjoin
x6
(
Sing
3
)
)
x5
|x5 ∈
x2
}
)
x4
=
binunion
(
binunion
x0
{
(
λ x6 .
SetAdjoin
x6
(
Sing
2
)
)
x5
|x5 ∈
x1
}
)
{
(
λ x6 .
SetAdjoin
x6
(
Sing
3
)
)
x5
|x5 ∈
x2
}
.
Apply binunion_idr with
binunion
(
binunion
x0
{
(
λ x4 .
SetAdjoin
x4
(
Sing
2
)
)
x3
|x3 ∈
x1
}
)
{
(
λ x4 .
SetAdjoin
x4
(
Sing
3
)
)
x3
|x3 ∈
x2
}
,
λ x3 x4 .
x4
=
binunion
(
binunion
x0
{
(
λ x6 .
SetAdjoin
x6
(
Sing
2
)
)
x5
|x5 ∈
x1
}
)
{
(
λ x6 .
SetAdjoin
x6
(
Sing
3
)
)
x5
|x5 ∈
x2
}
.
Let x3 of type
ι
→
ι
→
ο
be given.
Assume H0:
x3
(
binunion
(
binunion
x0
{
(
λ x5 .
SetAdjoin
x5
(
Sing
2
)
)
x4
|x4 ∈
x1
}
)
{
(
λ x5 .
SetAdjoin
x5
(
Sing
3
)
)
x4
|x4 ∈
x2
}
)
(
binunion
(
binunion
x0
{
(
λ x5 .
SetAdjoin
x5
(
Sing
2
)
)
x4
|x4 ∈
x1
}
)
{
(
λ x5 .
SetAdjoin
x5
(
Sing
3
)
)
x4
|x4 ∈
x2
}
)
.
The subproof is completed by applying H0.
■