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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
SNo
x0
.
Assume H1:
SNoLe
0
x0
.
Apply sqrt_SNo_nonneg_prop1a with
x0
leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x1 of type
ι
be given.
Assume H2:
x1
∈
SNoS_
(
SNoLev
x0
)
.
Assume H3:
SNoLe
0
x1
.
Apply SNoS_E2 with
SNoLev
x0
,
x1
,
and
(
and
(
SNo
(
sqrt_SNo_nonneg
x1
)
)
(
SNoLe
0
(
sqrt_SNo_nonneg
x1
)
)
)
(
mul_SNo
(
sqrt_SNo_nonneg
x1
)
(
sqrt_SNo_nonneg
x1
)
=
x1
)
leaving 3 subgoals.
Apply SNoLev_ordinal with
x0
.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Assume H4:
SNoLev
x1
∈
SNoLev
x0
.
Assume H5:
ordinal
(
SNoLev
x1
)
.
Assume H6:
SNo
x1
.
Assume H7:
SNo_
(
SNoLev
x1
)
x1
.
Apply sqrt_SNo_nonneg_prop1 with
x1
leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H3.
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