Let x0 of type ι be given.

Apply RealsStruct_Q_props with x0, RealsStruct_Q x0 ⊆ field0 x0 leaving 2 subgoals.

The subproof is completed by applying H0.

Assume H1: RealsStruct_Q x0 ⊆ field0 x0.

Assume H2: ∀ x1 . x1 ∈ RealsStruct_Q x0 ⟶ ∀ x2 : ο . (x1 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ RealsStruct_Z x0 ⟶ ∀ x4 . x4 ∈ RealsStruct_Npos x0 ⟶ field2b x0 x4 x1 = x3 ⟶ x2) ⟶ x2.

Assume H3: ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ RealsStruct_Z x0 ⟶ ∀ x3 . x3 ∈ RealsStruct_Npos x0 ⟶ field2b x0 x3 x1 = x2 ⟶ x1 ∈ RealsStruct_Q x0.

The subproof is completed by applying H1.

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Proofgold Proof