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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNoLe 0 x0.
Assume H3: SNoLe x0 x1.
Apply SNoLeE with x0, x1, SNoLe (sqrt_SNo_nonneg x0) (sqrt_SNo_nonneg x1) leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Assume H4: SNoLt x0 x1.
Apply SNoLtLe with sqrt_SNo_nonneg x0, sqrt_SNo_nonneg x1.
Apply sqrt_SNo_nonneg_mon_strict with x0, x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Assume H4: x0 = x1.
Apply H4 with λ x2 x3 . SNoLe (sqrt_SNo_nonneg x3) (sqrt_SNo_nonneg x1).
The subproof is completed by applying SNoLe_ref with sqrt_SNo_nonneg x1.
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