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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.
Assume H0: x0 ⊆ {x3 ∈ real|SNoLe 0 x3}.
Assume H1: x1 ⊆ {x3 ∈ real|SNoLe 0 x3}.
Assume H2: x2 ∈ real.
Assume H3: SNoLe 0 x2.
Let x3 of type ι be given.
Assume H4: x3 ∈ famunion x0 (λ x4 . {div_SNo (add_SNo x2 (mul_SNo x4 x5)) (add_SNo x4 x5)|x5 ∈ x1,SNoLt 0 (add_SNo x4 x5)}).
Apply SepI with real, SNoLe 0, x3 leaving 2 subgoals.
Apply SNo_sqrtauxset_real with x0, x1, x2, x3 leaving 4 subgoals.
Let x4 of type ι be given.
Assume H5: x4 ∈ x0.
Apply SepE1 with real, λ x5 . SNoLe 0 x5, x4.
Apply H0 with x4.
The subproof is completed by applying H5.
Let x4 of type ι be given.
Assume H5: x4 ∈ x1.
Apply SepE1 with real, λ x5 . SNoLe 0 x5, x4.
Apply H1 with x4.
The subproof is completed by applying H5.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Apply famunionE_impred with x0, λ x4 . {div_SNo (add_SNo x2 (mul_SNo x4 x5)) (add_SNo x4 x5)|x5 ∈ x1,SNoLt 0 (add_SNo x4 x5)}, x3, SNoLe 0 x3 leaving 2 subgoals.
The subproof is completed by applying H4.
Let x4 of type ι be given.
Assume H5: x4 ∈ x0.
Assume H6: x3 ∈ {div_SNo (add_SNo x2 (mul_SNo x4 x5)) (add_SNo x4 x5)|x5 ∈ x1,SNoLt 0 (add_SNo x4 x5)}.
Apply ReplSepE_impred with x1, λ x5 . SNoLt 0 (add_SNo x4 x5), λ x5 . div_SNo (add_SNo x2 (mul_SNo x4 x5)) (add_SNo x4 x5), x3, SNoLe 0 x3 leaving 2 subgoals.
The subproof is completed by applying H6.
Let x5 of type ι be given.
Assume H7: x5 ∈ x1.
Assume H8: SNoLt 0 (add_SNo x4 x5).
Assume H9: x3 = div_SNo (add_SNo x2 (mul_SNo x4 x5)) (add_SNo x4 x5).
Apply SepE with real, λ x6 . SNoLe 0 x6, x4, SNoLe 0 x3 leaving 2 subgoals.
Apply H0 with x4.
The subproof is completed by applying H5.
Assume H10: x4 ∈ real.
Assume H11: SNoLe 0 x4.
Claim L12: ...
...
Apply SepE with real, λ x6 . SNoLe 0 x6, x5, SNoLe 0 x3 leaving 2 subgoals.
Apply H1 with x5.
The subproof is completed by applying H7.
Assume H13: x5 ∈ real.
Assume H14: SNoLe 0 x5.
Claim L15: ...
...
Apply H9 with λ x6 x7 . SNoLe 0 x7.
Apply SNoLeE with 0, add_SNo x2 (mul_SNo x4 x5), SNoLe 0 (div_SNo (add_SNo x2 (mul_SNo x4 x5)) (add_SNo x4 x5)) leaving 5 subgoals.
The subproof is completed by applying SNo_0.
Apply SNo_add_SNo with x2, mul_SNo x4 x5 leaving 2 subgoals.
Apply real_SNo with x2.
The subproof is completed by applying H2.
Apply SNo_mul_SNo with x4, x5 leaving 2 subgoals.
The subproof is completed by applying L12.
The subproof is completed by applying L15.
Apply add_SNo_0R with 0, λ x6 x7 . SNoLe x6 (add_SNo x2 (mul_SNo x4 x5)) leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply add_SNo_Le3 with 0, 0, x2, mul_SNo x4 x5 leaving 6 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying SNo_0.
Apply real_SNo with x2.
The subproof is completed by applying H2.
Apply SNo_mul_SNo with x4, x5 leaving 2 subgoals.
The subproof is completed by applying L12.
The subproof is completed by applying L15.
The subproof is completed by applying H3.
Apply SNoLeE with 0, x4, SNoLe 0 (mul_SNo x4 x5) leaving 5 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying L12.
The subproof is completed by applying H11.
Assume H16: SNoLt 0 x4.
Apply SNoLeE with 0, x5, SNoLe 0 (mul_SNo x4 x5) leaving 5 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying L15.
The subproof is completed by applying H14.
Assume H17: SNoLt 0 ....
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