Beginning of Section A554
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use <= as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Variable F1 : setsetset
Hypothesis HF1 : ∀x0int, ∀x1int, F1 x0 x1 int
Variable G1 : setset
Hypothesis HG1 : ∀x0int, G1 x0 int
Variable H1 : setset
Hypothesis HH1 : ∀x0int, H1 x0 int
Variable U1 : setsetset
Hypothesis HU1 : ∀x0int, ∀x1int, U1 x0 x1 int
Variable V1 : setset
Hypothesis HV1 : ∀x0int, V1 x0 int
Variable F0 : setset
Hypothesis HF0 : ∀x0int, F0 x0 int
Variable G0 : set
Hypothesis HG0 : G0 int
Variable H0 : setset
Hypothesis HH0 : ∀x0int, H0 x0 int
Variable U0 : setsetset
Hypothesis HU0 : ∀x0int, ∀x1int, U0 x0 x1 int
Variable V0 : setset
Hypothesis HV0 : ∀x0int, V0 x0 int
Variable SMALL : setset
Hypothesis HSMALL : ∀x0int, SMALL x0 int
Variable F2 : setset
Hypothesis HF2 : ∀x0int, F2 x0 int
Variable G2 : setset
Hypothesis HG2 : ∀x0int, G2 x0 int
Variable H2 : set
Hypothesis HH2 : H2 int
Variable U2 : setsetset
Hypothesis HU2 : ∀x0int, ∀x1int, U2 x0 x1 int
Variable V2 : setset
Hypothesis HV2 : ∀x0int, V2 x0 int
Variable FAST : setset
Hypothesis HFAST : ∀x0int, FAST x0 int
Hypothesis H1 : (∀Xint, (∀Yint, ((F1 X Y) = ((X + X) + Y))))
Hypothesis H2 : (∀Xint, ((G1 X) = X))
Hypothesis H3 : (∀Xint, ((H1 X) = X))
Hypothesis H4 : (∀Xint, (∀Yint, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
Hypothesis H5 : (∀Xint, ((V1 X) = (U1 (G1 X) (H1 X))))
Hypothesis H6 : (∀Xint, ((F0 X) = ((1 + (2 + X)) * (V1 X))))
Hypothesis H7 : (G0 = 1)
Hypothesis H8 : (∀Xint, ((H0 X) = (1 + X)))
Hypothesis H9 : (∀Xint, (∀Yint, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
Hypothesis H10 : (∀Xint, ((V0 X) = (U0 G0 (H0 X))))
Hypothesis H11 : (∀Xint, ((SMALL X) = (V0 X)))
Hypothesis H12 : (∀Xint, ((F2 X) = (X + X)))
Hypothesis H13 : (∀Xint, ((G2 X) = X))
Hypothesis H14 : (H2 = 2)
Hypothesis H15 : (∀Xint, (∀Yint, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
Hypothesis H16 : (∀Xint, ((V2 X) = (U2 (G2 X) H2)))
Hypothesis H17 : (∀Xint, ((FAST X) = (((2 + (2 + X)) * ((V2 X) + - 1)) * (1 + (2 + X)))))
Theorem. (A554)
(∀Nint, ((0 <= N)((SMALL N) = (FAST N))))
Proof:
The rest of the proof is missing.

End of Section A554