\chapter{Surreal Numbers II}
\section{SurrealArithmetic}\label{dffhchchfgbgmgbechjgehigngfgehjgdg}\begin{definition}\label{def:ngjgogfhdhpfdfoepg}
$\mname{minus\_SNo}$ is the opaque object of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 268a6c1da15b8fe97d37be85147bc7767b27098cdae193faac127195e8824808 \\
Pure Object Address: & TMcxvtkwUHDLCnnXkVE7rvLpYFkkCr9t57h \\
Theory Object Id: & 8a6d8513b8fae7a11b1033553a028661380ee0d23337db3b5de7c37c00bf7d3a \\
Theory Object Address: & TMNRx3zLagXCf6MQrfwBvyPXLa8o1ZnJoqf
\end{tabular}
\end{center}

\end{definition}
{\bf{Notation.}} We use $\prenc{-}$ as a prefix operator  corresponding to applying term ${\mname{minus\_SNo}}$.
{\bf{Notation.}} We use $\infmdnd$ as an infix operator  corresponding to applying term ${\mname{SNoLe}}$.
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpffgbh} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{\prenc{{x}}}={{{\mname{SNoCut}}~{\{{\prenc{{z}}}|z \in {{\mname{SNoR}}~{x}}\}}}~{\{{\prenc{{w}}}|w \in {{\mname{SNoL}}~{x}}\}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c9793e4a852e2a5d73a3008d3c0a88124864f04508e5f6cd84920975e8bddfc6 \\
Pure Prop Address: & TMFADt2W5uMVy5FYrgHvEE2vrdFR8Gnxb1b \\
Theory Prop Id: & 570cba2e3229759ec994c333f1852d951e44a44817d76474c936f0b2783d17fb \\
Theory Prop Address: & TMQTpg4CNj1EB19CWqcr8oBSbG7cvoVZJuJ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfahchpgahbd} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{{{\mname{SNo}}~{({\prenc{{x}}})}}\unicdcdcdhd{({\unicdcdadad u \in {{\mname{SNoL}}~{x}} .  {{\prenc{{x}}}<{\prenc{{u}}}}})}}\unicdcdcdhd{({\unicdcdadad u \in {{\mname{SNoR}}~{x}} .  {{\prenc{{u}}}<{\prenc{{x}}}}})}}\unicdcdcdhd{{{\mname{SNoCutP}}~{\{{\prenc{{z}}}|z \in {{\mname{SNoR}}~{x}}\}}}~{\{{\prenc{{w}}}|w \in {{\mname{SNoL}}~{x}}\}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7c0e82df70ecda3be7a8153a63670c1b82bc354a3b3bb658b240db5bf6079ccc \\
Pure Prop Address: & TMPFDUxTsU6Hgp2dqtAfHiUMzQdZC4s6Rpo \\
Theory Prop Id: & f0b75f0c13ab8217a5ebfacdc1aebc7bacaf4f0bcf65609bd5bdfc26f85fe25e \\
Theory Prop Address: & TMHp3P8GQ2PdxPN38fbjih9CXrrvAVxARmW
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfngjgogfhdhpfdfoepg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{\mname{SNo}}~{({\prenc{{x}}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5cf7d9b9d43ef2ee9370e1aa6dded6d26ae3d971cbc3b561b6d07723f8215269 \\
Pure Prop Address: & TMdc5VANKtJFyfviMQyb75gDTg48GudNzRZ \\
Theory Prop Id: & 744cc2ee9e53d674ba772a0aa685c56eb8e460e86e64854f7ce0be53b0da5fff \\
Theory Prop Address: & TMQ1ogbeVfSeNvyG9vTPaMVZg2dqUSJyjSo
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfmeehpfdgpgogehchbg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}<{y}}\unicdbdjdcd{{\prenc{{y}}}<{\prenc{{x}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5ce52d7493ef6b5572b747bf671cea2e6e6f80ceb28c95e2123521dd504b0b0f \\
Pure Prop Address: & TMYLjCwrhRZgQN5K5EwHjSFpA4jVNLAHCjn \\
Theory Prop Id: & 3ebf6e39ebb99f5ddf5eee4fb81a248a684fd1274696b9741a23625f9df6dc52 \\
Theory Prop Address: & TMPCextugz7QEf4NpPTNxs5QRszhndWEXZL
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfmefgpfdgpgogehchbg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}\unicdcdgded{y}}\unicdbdjdcd{{\prenc{{y}}}\unicdcdgded{\prenc{{x}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c1b2259be1bdf45a1f93fe5cc80752b95adc141e37aea0fc5974e675529dbab0 \\
Pure Prop Address: & TMSdtEKoTTw4VsgYxJhGDbFrsEav7bgzwDz \\
Theory Prop Id: & f8c8964a3e52fb3b06f65d0148f443ef72cb1a1710977f309628d62a6dc762ff \\
Theory Prop Address: & TMR6iQfy2ketSGqusuoDGBAoZDHFdKZi5ei
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfdfoepgdefhehaf} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNoCutP}}~{\{{\prenc{{z}}}|z \in {{\mname{SNoR}}~{x}}\}}}~{\{{\prenc{{w}}}|w \in {{\mname{SNoL}}~{x}}\}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 99c6fcde5ba109860366fc8eb324200f6b134a75dc50371fa98ecbf13fe8dd9f \\
Pure Prop Address: & TMPGzP3WvAeSXVX4PkvAXm9hswTwYn7Lx5a \\
Theory Prop Id: & 5ad5e46892ec6dde1a8e632b515b72dd9624c207d92d7f5fc1b887ec45170a91 \\
Theory Prop Address: & TMYA5Ns3A1StK4eq1kfygXeX7mudw4n946H
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfdfoepgdefhehafpfhgfgog} ${\unicdcdadad L R .  {{{{\mname{SNoCutP}}~{L}}~{R}}\unicdbdjdcd{{{\mname{SNoCutP}}~{\{{\prenc{{z}}}|z \in {R}\}}}~{\{{\prenc{{w}}}|w \in {L}\}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e7650d8c5c9fb96894ef4334966c9c4043dc7b919ef49e218164db5101a12296 \\
Pure Prop Address: & TMcYQN83fYBS39M2EEEbDtftNsYeWHCkMAi \\
Theory Prop Id: & 687fea2cc86a5e6dc4aaff906beb9303fa12e95620a78db83430a8be9dfd1bfc \\
Theory Prop Address: & TMK1WeYsUqdVKtQ7kU3SZV4h8vJvGUyosuH
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfmefgghpfmgfgngbd} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad x \in {{\mname{SNoS\_}}~{\alpha}} .  {{{\mname{SNoLev}}~{({\prenc{{x}}})}} \subseteq {{\mname{SNoLev}}~{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 14986a810fa991813a940caff72106cdeb17e367c8dc9c6eeb88ab77ed7fb70c \\
Pure Prop Address: & TMGTTrCpJEdP2JL7ZKjHkPvFjc3gfmdeYX5 \\
Theory Prop Id: & afc115b1b5f9ff0b6226e6d37a5708b45bf273bc776bf7816207943b0cf564fa \\
Theory Prop Address: & TMcrjfrThFM8M6pTTZtnJdFZBn72x9XkSzf
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfmefgghpfmgfgngcd} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNoLev}}~{({\prenc{{x}}})}} \subseteq {{\mname{SNoLev}}~{x}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 87902034fac50c9d369be3b80426e4d2a737cf936787a4c0f6c8ad74e0843878 \\
Pure Prop Address: & TMbVFKSGdQJSr2kHyJUmNCjt7CTD1D6JKHf \\
Theory Prop Id: & 19cf80ac01fc15c5bac2a1a546c9cfca35cf8eb2a455b3cf3fb66a2a7fe614f8 \\
Theory Prop Address: & TMQtKJ6HCGKKXSHeLSJwRyRPcKmC4ZiLZK5
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfjgogghpgmg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{\prenc{{\prenc{{x}}}}}={x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 44bfe89261da61b19b9a5cd41ede8430faa107c7822701da47ef38cb4549faa6 \\
Pure Prop Address: & TMc77Sp2dpzatNwekdynQs9vhSPfJgGXk1U \\
Theory Prop Id: & cc64eeea172423c51379afadb3e01de9b7f2513ce938649ddcb32805f5bdafb0 \\
Theory Prop Address: & TMTt3PRTf9CD5ikrtEqSt2QevQVtKjQrFbz
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfmefggh} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNoLev}}~{({\prenc{{x}}})}}={{\mname{SNoLev}}~{x}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f65499af9040b6b3e15497cab281d7675bff566bb6d4fdd182cc0343568ae07e \\
Pure Prop Address: & TMHMPna5msJBVV2aj6dRhmCueZSUBa2VhT5 \\
Theory Prop Id: & 476ae5e0d40774e983e1d64325be7c212ae84435e983e476cb99925933ecd248 \\
Theory Prop Address: & TMRdSSgdRywiPXPvi9bXyTwx2JCnJRrLkHS
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfdfoepgpf} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad x .  {{{{\mname{SNo\_}}~{\alpha}}~{x}}\unicdbdjdcd{{{\mname{SNo\_}}~{\alpha}}~{({\prenc{{x}}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 84605fa359f5f8a80873f561cdcf5e371180d9965ea1157048ba66f9af585e90 \\
Pure Prop Address: & TMZviY6yHgD2P1xafVTUWcBcjX291SfHuZq \\
Theory Prop Id: & 14bd6d1aecca30a4c446e92cca90fa66fd7b0499acca03daf8cfa848740f428f \\
Theory Prop Address: & TMTsLyGVKuoMiH43ymXqPkKMWYDN3HyfXHd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfdfoepgdfpf} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad x .  {{{x} \in {{\mname{SNoS\_}}~{\alpha}}}\unicdbdjdcd{{\prenc{{x}}} \in {{\mname{SNoS\_}}~{\alpha}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 12f1a5194619e18f763ea1432c60862d2c280bcc57c6b538e7af0b8563b3aef0 \\
Pure Prop Address: & TMNu6dbAKA4NtYuwXCDn3Z2xwPtwvtj71gH \\
Theory Prop Id: & e10c92f39791767985ef5b9f5eed352a930cae37c683dfacbd6f47c7d2d23e93 \\
Theory Prop Address: & TMVtCJL3NR1TVjUhDpc8LMfGugbda1XLCG2
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgdefhehpffgbhpfmgfgng} ${\unicdcdadad v .  {{{\mname{SNo}}~{v}}\unicdbdjdcd{\unicdcdadad L R .  {{{{\mname{SNoCutP}}~{L}}~{R}}\unicdbdjdcd{{{v}={{{\mname{SNoCut}}~{L}}~{R}}}\unicdbdjdcd{{\prenc{{v}}}={{{\mname{SNoCut}}~{\{{\prenc{{z}}}|z \in {R}\}}}~{\{{\prenc{{w}}}|w \in {L}\}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 29069a9700032f41c27cbce5d097d02ca500af7402feea86befd364a812bea18 \\
Pure Prop Address: & TMJJorcHcCvbte94Wk2mohN3ofdRRZi9USf \\
Theory Prop Id: & 11eb03174b07f424338800b6644cd5921561dc680adb86b22f9254abda306195 \\
Theory Prop Address: & TMRBupH2YERv2pxoxNHvbLaoKkUEWp3LmVu
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgdefhehpffgbh} ${\unicdcdadad L R .  {{{{\mname{SNoCutP}}~{L}}~{R}}\unicdbdjdcd{{\prenc{{{{\mname{SNoCut}}~{L}}~{R}}}}={{{\mname{SNoCut}}~{\{{\prenc{{z}}}|z \in {R}\}}}~{\{{\prenc{{w}}}|w \in {L}\}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 083ffe4c6522008e61c2b89e0e64c5db7b371419a23a3f14a519e0582688fb5b \\
Pure Prop Address: & TMaD6A1fin9xGjs9gJk55ba3XE4iuTh2W7Y \\
Theory Prop Id: & 54736f85eb44e8ebcd3e07eea3d33c79842c2ce605f7a6384f4ba14b9eec7d07 \\
Theory Prop Address: & TMGrpE8HLXgUszxGm966iTpad2EhS6XSckh
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfmeehpfdgpgogehchbgbd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\prenc{{x}}}<{y}}\unicdbdjdcd{{\prenc{{y}}}<{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 226757cb052a4956017abe14687f3e1caa99a28fde325c1d779946bba7658526 \\
Pure Prop Address: & TMXXUYTV6XCxwBUgA7MgZTzj6H9ahkRPPTo \\
Theory Prop Id: & 29872961c38defbc42168eadfbb395ff6b34bb794dfd9e81bcce6ad6374039e9 \\
Theory Prop Address: & TMEs4ZMsQaGiy3xaRcw9TpZk6KCqk5ye83u
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfmeehpfdgpgogehchbgcd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}<{\prenc{{y}}}}\unicdbdjdcd{{y}<{\prenc{{x}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c6f2dabf5db1946ff00666f185293c94dc027bfbc3c6b34ab842300e6fc61e10 \\
Pure Prop Address: & TMK1aCcPBp5isVNhukJBv5WwEVryhWR39im \\
Theory Prop Id: & f419684217fb1e0747adc0f778ba066f701845079345dd8aaa40b5499c5cd992 \\
Theory Prop Address: & TMEu6PgqrG75bbqAH6z2eo8taFDwoWi72LL
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfmeehpfdgpgogehchbgdd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\prenc{{x}}}<{\prenc{{y}}}}\unicdbdjdcd{{y}<{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 30b6d6ce0519f672dafb8da416d1f08cc773f7718ea0c0e59fe2b674a8a28eb5 \\
Pure Prop Address: & TMEiiHLSUJb67TTgfxay254xNKWCum3TL6U \\
Theory Prop Id: & 20ebc3b1dcf4268fcbbda61e77e0b1aa0cd9de3c07e995cd027e1623022ab3a6 \\
Theory Prop Address: & TMdv9HV4i4ia6bWB1AEuc8hYPUVNAMAF4i3
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfad} ${{\prenc{{0}}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 77bc6707cfcc559fd1f9509caeb31aaec5c6ea27e99983f5ed2a0b31aba16371 \\
Pure Prop Address: & TMExypvZyPTdXH5VP42X3hVoCRLY5xqMGSS \\
Theory Prop Id: & 3fcc57eb71cf1d99ef0ab04ed19f5bc54b596714fb2b6d70ce2f61b05445433c \\
Theory Prop Address: & TMS4JbQtTa5bFrxAcTD38DVaprpkd1UddT2
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfngpgngfghgbg} ${{\mname{SNo}}~{({\prenc{{\mname{omega}}}})}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9682bc4f449aab7850dab8b061240c04bbfa925a9af0c6d436b57571e88bf054 \\
Pure Prop Address: & TMbs8VnGZaJLV9PMfUU3ZA9UMWppyWZCbpk \\
Theory Prop Id: & e72fb1696565b219d098fc491769ed67f15f1c716f9ba159513fa19c4a3cfb68 \\
Theory Prop Address: & TMQGDzKedUx3YyBnazoAQzc2BJYMagTe9aU
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngpgchegjgogbgmgpfdfoepg} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{SNo}}~{({\prenc{{\alpha}}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 67157695184d77aa292d2e91ae058aab5728c1aa262ad91430d98e01a61ecee5 \\
Pure Prop Address: & TMZCauMQgviQfBsquF7s2dXY5ovy2NZhJmB \\
Theory Prop Id: & 533947d00d9f2f544511c3e8876c2d3bec39a85cf5de0507579619af91b1cfe3 \\
Theory Prop Address: & TMdpAbWGcEmUBU6mDwF1QUJ8J3YQKqAnC2W
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngpgchegjgogbgmgpfdfoepgmefggh} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{SNoLev}}~{({\prenc{{\alpha}}})}}={\alpha}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 872283abfa6f0dfbb1d72e382868b2cf0992ece74aabc0727d30681dd1c82466 \\
Pure Prop Address: & TMWf88dibCFMEwQnUShPjR7f5qzRTHihxjA \\
Theory Prop Id: & 425f79bccb3002745cf4cfe3c4ee8e5f28fc81700abf759ffe8688c3ec559879 \\
Theory Prop Address: & TMJYCVXHmoi2fzEKN7sFk63txQj7WCuRVYU
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngpgchegjgogbgmgpfdfoepgmefgghpfngjgog} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad z .  {{{\mname{SNo}}~{z}}\unicdbdjdcd{{{{\mname{SNoLev}}~{z}} \in {\alpha}}\unicdbdjdcd{{\prenc{{\alpha}}}<{z}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c28e6646c6374a60e623edaf3e98581ff82a2d308bc362c7ed683cb7ef763c9b \\
Pure Prop Address: & TMGx6jRVew4w8A8QL13WubtT3gsqgBkjuKj \\
Theory Prop Id: & a11cfa7f908447229087de2b3b5ed724471c04c5ace74fef64304120331db0b2 \\
Theory Prop Address: & TMcMHpMnYb8FFYn3kS1X4hRDwLCnUBPrPN2
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngpgchegjgogbgmgpfdfoepgmefgghpfngjgogpfcd} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad z .  {{{\mname{SNo}}~{z}}\unicdbdjdcd{{{{\mname{SNoLev}}~{z}} \in {{\mname{ordsucc}}~{\alpha}}}\unicdbdjdcd{{\prenc{{\alpha}}}\unicdcdgded{z}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2766c7ac03ddae43cd989f364bd3933e33016953b97822589263b8813a782baf \\
Pure Prop Address: & TMdLNMofLk9MRXKxrVStAmXcnfLmpPf5ZLD \\
Theory Prop Id: & a96d68eb40e4505dff800c19d3a67d927597a15491c19c2e7613c3653faa7592 \\
Theory Prop Address: & TMUbQC9sotj3Sdmk1rPhzFc2uGHxwQ7fTKG
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfdfoepgdfpfpgngfghgbg} ${\unicdcdadad x \in {{\mname{SNoS\_}}~{\mname{omega}}} .  {{\prenc{{x}}} \in {{\mname{SNoS\_}}~{\mname{omega}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & daeba565dddc519103825c78e55714d95a0ae94548ea309e8d64ab2518ce1c95 \\
Pure Prop Address: & TMHUVaZspynr2H3jcHXqWoRHirGbNS7NfFb \\
Theory Prop Id: & 0c19f5cfaa9b340bbf0166314f1046d3ad7c8558b64304949986f6ee62b7946e \\
Theory Prop Address: & TMV7G2rh3rEAwiDk5QRMvAdQX4xEJpR6g26
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:bgegegpfdfoepg}
$\mname{add\_SNo}$ is the opaque object of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 127d043261bd13d57aaeb99e7d2c02cae2bd0698c0d689b03e69f1ac89b3c2c6 \\
Pure Object Address: & TMHBQoir8zBmsd18ceCptzsSPneX7e796Qh \\
Theory Object Id: & f62f9ad0a4b78ce9ea1f70abc3f81bb4efd2f55019f5ea81fec6bff7d056a9c4 \\
Theory Object Address: & TMLfAbLP1JzEe3VPaGmScRvpGeCCLs2hBkM
\end{tabular}
\end{center}

\end{definition}
{\bf{Notation.}} We use $\inflc$ as a right associative infix operator  corresponding to applying term ${\mname{add\_SNo}}$.
\begin{theorem}\label{thm:bgegegpfdfoepgpffgbh} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{\unicdcdadad y .  {{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}+{y}}={{{\mname{SNoCut}}~{({{\{{{w}+{y}}|w \in {{\mname{SNoL}}~{x}}\}}\unicdcdcdbg{\{{{x}+{w}}|w \in {{\mname{SNoL}}~{y}}\}}})}}~{({{\{{{z}+{y}}|z \in {{\mname{SNoR}}~{x}}\}}\unicdcdcdbg{\{{{x}+{z}}|z \in {{\mname{SNoR}}~{y}}\}}})}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0c11282e349ac03ead817fec6965779c38db2bba16e6617397d7a00fec0025f2 \\
Pure Prop Address: & TMUsqiTidRPvWsTgAU6Pe8thY6esLYajg8X \\
Theory Prop Id: & 437a9c143d85d0cbb514a028f14f083cc0f8d79aa47f3997255c796243fc0fe7 \\
Theory Prop Address: & TMGPPTS21hFunW6FDmNfEs1RLtzqupLkwJY
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfahchpgahbd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{{{{{\mname{SNo}}~{({{x}+{y}})}}\unicdcdcdhd{({\unicdcdadad u \in {{\mname{SNoL}}~{x}} .  {{{u}+{y}}<{{x}+{y}}}})}}\unicdcdcdhd{({\unicdcdadad u \in {{\mname{SNoR}}~{x}} .  {{{x}+{y}}<{{u}+{y}}}})}}\unicdcdcdhd{({\unicdcdadad u \in {{\mname{SNoL}}~{y}} .  {{{x}+{u}}<{{x}+{y}}}})}}\unicdcdcdhd{({\unicdcdadad u \in {{\mname{SNoR}}~{y}} .  {{{x}+{y}}<{{x}+{u}}}})}}\unicdcdcdhd{{{\mname{SNoCutP}}~{({{\{{{w}+{y}}|w \in {{\mname{SNoL}}~{x}}\}}\unicdcdcdbg{\{{{x}+{w}}|w \in {{\mname{SNoL}}~{y}}\}}})}}~{({{\{{{z}+{y}}|z \in {{\mname{SNoR}}~{x}}\}}\unicdcdcdbg{\{{{x}+{z}}|z \in {{\mname{SNoR}}~{y}}\}}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f3ece7fc5faeb9b5356d40372aef6e582964fa486d2e44e7f3eed3f05d52e3e0 \\
Pure Prop Address: & TMcpTrmdaf7f5xyc6e4BpSWAmxNsmf3E5yS \\
Theory Prop Id: & 5a879f7365529b11a3ffce943b27ada13bc1e46e2af4e2258090e38d55606b7f \\
Theory Prop Address: & TMRNpdhJ6PXJ334PSwBu4u3N4MRpHafuGvT
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfbgegegpfdfoepg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{\mname{SNo}}~{({{x}+{y}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c4a9a6e12a8ec5b5bebaa97ea0d920bd43eea74d9b3d0ffda15762cf19565969 \\
Pure Prop Address: & TMPtcdp9k9YMXhBQVaR4jVorJnSfZBVvKki \\
Theory Prop Id: & a57095a9caa997f445ebde562cde8dd04d8e8a5fd2a57d1c3a9f8decf60ccdeb \\
Theory Prop Address: & TMXsdqtKxXjtQCrsnZQNbBCQsbgVDjWXpMf
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfbgegegpfdfoepgpfdd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{\mname{SNo}}~{({{x}+{{y}+{z}}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f6ef07d3f7fb2565f3fe4732b883d4499d8442b69d6ae7d977010d18d4d54ecf \\
Pure Prop Address: & TMZ4dtBp9Qizyu4nFxn7a2N9SQRVoE9Q4W8 \\
Theory Prop Id: & f2d78c7d8aa6405036ae26879362dfde9af9a15cdbbf11ba2bff0d754b7750ca \\
Theory Prop Address: & TMVox3RUbkj9ZYffbdvRh7znGFXNfLVwPFV
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfbgegegpfdfoepgpfed} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{\mname{SNo}}~{({{x}+{{y}+{{z}+{w}}}})}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d71f0eb42650c644bb0c4a2fbb29173cc3bf47fea9dd9401c0ba715333db1820 \\
Pure Prop Address: & TMF4oHEkLVa8KD2NcVmcFum9JhucpudfEb1 \\
Theory Prop Id: & 42037e0c85566de52aba1c0f97268637a493ef459b07be2723bfca5dc7b13a97 \\
Theory Prop Address: & TMXXq7JGxapVLMuBsmWzgDWpgqW72RtAMLV
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehbd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}<{z}}\unicdbdjdcd{{{x}+{y}}<{{z}+{y}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 92232f18c39390c575ed22b7c92927a25b5ade867b5e2ed5f2cdf7e509909805 \\
Pure Prop Address: & TMLEq8tneuw32G568W1mBtwRTNNgQPCRYyz \\
Theory Prop Id: & cb0738c7872b30b23cb65bbcd1ce88351f2466b21c1b5cc15f5b44ae581e28e0 \\
Theory Prop Address: & TMWTBYY2jenWVwUwvSromB8AgRWDVcJnbY6
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmefgbd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}\unicdcdgded{z}}\unicdbdjdcd{{{x}+{y}}\unicdcdgded{{z}+{y}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d8f37ed7006da1a1793317440a31b24b6f87f27cad9329c38e8a7ff9753d2dcc \\
Pure Prop Address: & TMaVRrrSjDXkuboEtM4VK6QYgB6UVbfBcaG \\
Theory Prop Id: & abcf7263a107c501b54d30826fb67a71bfd65aa0e1fa33d5cb94ba28bbd40502 \\
Theory Prop Address: & TMR5TeFwaZGZUnBsDVqT64ZqwgMewHGgCDF
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehcd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{y}<{z}}\unicdbdjdcd{{{x}+{y}}<{{x}+{z}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6ad46463d6f79423eff3ac9a3ac6201e44983b342ffb704d9cde5795efc8a7ec \\
Pure Prop Address: & TMZabfrosx2WbihjxmG74tgmZaLS7LCKpAU \\
Theory Prop Id: & 6f0941dc05860da5ece42171247edca54127ab818e70661d52995088db287957 \\
Theory Prop Address: & TMQWhSEEumkynNocoC5sSjUxscps15Jp64G
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmefgcd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{y}\unicdcdgded{z}}\unicdbdjdcd{{{x}+{y}}\unicdcdgded{{x}+{z}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 14975eae6645de30652349b64a88f0b2e35b84bc21be97aafc239516dd628b87 \\
Pure Prop Address: & TMdukf7SbuQ39yuQq8KCjup31RNEDuTLQnT \\
Theory Prop Id: & 3f52ba378d30a817615bc0f6bf9697c00b83ce446cbe3de8fff3a6d92b6bd5de \\
Theory Prop Address: & TMGdyU2zNNkb5T6LeAwxX74oWd2P7WYJJ8c
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehddbg} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{x}<{z}}\unicdbdjdcd{{{y}\unicdcdgded{w}}\unicdbdjdcd{{{x}+{y}}<{{z}+{w}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b2215f63878ba2a6fc09f12939fd6fc51dc580691897e8323370abb12c70f7bd \\
Pure Prop Address: & TMYyEHQwicHmVL6MUyPu2QNJHpVu8pL5YVP \\
Theory Prop Id: & 18ff91f30d21406ebb6b75675c0b1f7e8ad283cbf98775f53dd3e656a5fa522f \\
Theory Prop Address: & TMSmpE2jeB2cyTEuG7Rq8sykhPLAACRxekP
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehddcg} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{x}\unicdcdgded{z}}\unicdbdjdcd{{{y}<{w}}\unicdbdjdcd{{{x}+{y}}<{{z}+{w}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5b4f92cfc6a3acaa35dd069d515cd4c025daa3b2bca2a7add6810dc336b0235f \\
Pure Prop Address: & TMKXJb12jTxUrEE48Fsz3YxLGNM4wfBPXJW \\
Theory Prop Id: & 8fe0d98eaddd0798a52530dbd10ab1d00f92e187620a37fb685f5830bd4c085c \\
Theory Prop Address: & TMKS8vtMN5qWoxjxHv7NEEavnk6emGyiiMz
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehdd} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{x}<{z}}\unicdbdjdcd{{{y}<{w}}\unicdbdjdcd{{{x}+{y}}<{{z}+{w}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7141a86b460a8ecf3af113551c387396eea76973c00c0ae6d3a7a9f0806449bd \\
Pure Prop Address: & TMNriuSM92c9RucKEto1JSRuMZvj8bjdnr5 \\
Theory Prop Id: & 55bdabe93657d73da3016bb38681d52b49a61dca9be7e4b0f99461db17aaf692 \\
Theory Prop Address: & TMKtJcmnivzZgTkMMwdV5vUEUxKGyMkrQFd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmefgdd} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{x}\unicdcdgded{z}}\unicdbdjdcd{{{y}\unicdcdgded{w}}\unicdbdjdcd{{{x}+{y}}\unicdcdgded{{z}+{w}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0be637ad324a3ea9caae189e47f4890cba93a45121b22ff1e10fad675f87f12c \\
Pure Prop Address: & TMSVKSkQGVqbboR9prtEBre18xSpVMapMrK \\
Theory Prop Id: & be66ca7a659e973fad15e883dbd8397986cfe73bd862e7b50ccf70a912a6e0b0 \\
Theory Prop Address: & TMdD3NhDM7mVSeEXJrP8Vh8HeE3of9UjkEu
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdfoepgdefhehaf} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNoCutP}}~{({{\{{{w}+{y}}|w \in {{\mname{SNoL}}~{x}}\}}\unicdcdcdbg{\{{{x}+{w}}|w \in {{\mname{SNoL}}~{y}}\}}})}}~{({{\{{{z}+{y}}|z \in {{\mname{SNoR}}~{x}}\}}\unicdcdcdbg{\{{{x}+{z}}|z \in {{\mname{SNoR}}~{y}}\}}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e2b7d74c79711fae6a14f4d67a701081855be3bb26fd923a62a391e04160bfae \\
Pure Prop Address: & TMdKu7mpdNdzCSmXhFYuNuBjrmEsPPsqqGm \\
Theory Prop Id: & f86b1a41eabc3959fd8be49bbc8865a0009802c350299879c6b3a1eb97530c7d \\
Theory Prop Address: & TMR6c36epFgVd323wsqJDSNi9GvJ5dHX5bG
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdfoepgdefhehafpfhgfgog} ${\unicdcdadad Lx Rx Ly Ry .  {{{{\mname{SNoCutP}}~{Lx}}~{Rx}}\unicdbdjdcd{{{{\mname{SNoCutP}}~{Ly}}~{Ry}}\unicdbdjdcd{{{\mname{SNoCutP}}~{({{\{{{w}+{{{\mname{SNoCut}}~{Ly}}~{Ry}}}|w \in {Lx}\}}\unicdcdcdbg{\{{{{{\mname{SNoCut}}~{Lx}}~{Rx}}+{w}}|w \in {Ly}\}}})}}~{({{\{{{z}+{{{\mname{SNoCut}}~{Ly}}~{Ry}}}|z \in {Rx}\}}\unicdcdcdbg{\{{{{{\mname{SNoCut}}~{Lx}}~{Rx}}+{z}}|z \in {Ry}\}}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 075eaabfcc7432298a2db5f87aafbc0106915ca041c774d445bb94d0821d5824 \\
Pure Prop Address: & TMS5jpucdv4wNPKAJszA5iBP9k5qK9fGjf2 \\
Theory Prop Id: & 9afab2f26e1613aa776c9706130a926c7e2da1ffe974ae51ddb5a8b57c102cf0 \\
Theory Prop Address: & TMUA1C1f53xTBiuMvrMGVhLK8tVv3uALH6H
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdgpgng} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}+{y}}={{y}+{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 548814e9730b34ca53c6149a1239b003d299813e4e1610c765a8f32771db006e \\
Pure Prop Address: & TMb6qo68fsewkpEkdhHBPyCDbKiaep9rQ9B \\
Theory Prop Id: & f017e50fae3edfeeaedc58af7e2473b3c3008dd0d65d3dd902ce9902cfcedaed \\
Theory Prop Address: & TMdQusEuxAaq4pjYGPyZyTsGeCDoe7ixEg5
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfadme} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{0}+{x}}={x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 107598652879bcb8baa57261d6be92d58926c75b0158561c7a0fbf824b5533ac \\
Pure Prop Address: & TMPpszvpgSas6jmAy2oPhZkMYNxdnAa9bCr \\
Theory Prop Id: & 446bada3d91ea404046f63fd5ef3b884131bc7c36182164979919c7306a98104 \\
Theory Prop Address: & TMLtAihTfZ4tMgum6Ca1thK1rJp885zUT7m
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfadcf} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{x}+{0}}={x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d8a186fe010013b2755c30ba0a202ddabfa4c545c10aac8b4e1364cac61cebd6 \\
Pure Prop Address: & TMTawHrBTSYsj41yX9x5qiLExuLJBg499YB \\
Theory Prop Id: & 1a7fa1896336c0b5037e49fd8da8f614bd60fec53a52508d1c24eb4c32d2c081 \\
Theory Prop Address: & TMMmjuw778ta6y4jTCoCMS5fmu1gZ8B6AGF
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfdfoepgpfmgjgoggh} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\prenc{{x}}}+{x}}={0}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 42471917566847dbc969b69ad8eb5cfbb677e19d7005aaefa3ddbcd227fbc72c \\
Pure Prop Address: & TMPWBgm88UpCF22WujXVVkVtCu8iciY3QTW \\
Theory Prop Id: & 44e251b28aec6132ef7e2d0a0c34be7ac5c327d1039a329806f84dd65df681f5 \\
Theory Prop Address: & TML2RWMKucEcfgS9pHC3eNVJ6jgXjXCWKn2
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfbgegegpfdfoepgpfdddg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{\mname{SNo}}~{({{x}+{{y}+{\prenc{{z}}}}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 104eb97ceeb39012baf2633f2295fcd8488c5da863f883a4f0bd3508d7022b82 \\
Pure Prop Address: & TMVNoU8QDePUm51AQC5k9QeJwu735iBxPvj \\
Theory Prop Id: & 833be87302ff13765fe452b02d5b463172269368fd9039c3342e653a13738d59 \\
Theory Prop Address: & TMPvYhSo4MBSTDX3QY6YVnoEkB94k85UJj6
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfdfoepgpfchjgoggh} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{x}+{\prenc{{x}}}}={0}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9e8c44fbf607e617cb13942672b1aaa08da477c7c043bd3445e12df7880842e3 \\
Pure Prop Address: & TMYMYkykBhLq1apDzVyC2vJc6hiCZ65cEJD \\
Theory Prop Id: & b2f9cf9ffdd19a14a7d49bd3eecd70f6ba950f3bae905926575b58dbe69c5611 \\
Theory Prop Address: & TMS4a619pMmyZvGY8DXex1wWCwRRFpTtwJ7
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfpgchegjgogbgmgpfdfoepgdefhehaf} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad beta .  {{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{{\mname{SNoCutP}}~{({{\{{{x}+{\beta}}|x \in {{\mname{SNoS\_}}~{\alpha}}\}}\unicdcdcdbg{\{{{\alpha}+{x}}|x \in {{\mname{SNoS\_}}~{\beta}}\}}})}}~{\mname{Empty}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2823e439cb46c064666c0dd60fea0fc706e8f3518725e3e71e9fe27292df89a0 \\
Pure Prop Address: & TMUPaueAJ3SYX5RiMrtRmYAsP3Gv5ibyiJr \\
Theory Prop Id: & 507eab0935b7feec7b7116c01bd1ac27c74326982b1b48e41b6c44b87990749b \\
Theory Prop Address: & TMT1Gz8K4EguVmo5NB2Woi7sw5sQBm85KE2
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfpgchegjgogbgmgpffgbh} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad beta .  {{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{{\alpha}+{\beta}}={{{\mname{SNoCut}}~{({{\{{{x}+{\beta}}|x \in {{\mname{SNoS\_}}~{\alpha}}\}}\unicdcdcdbg{\{{{\alpha}+{x}}|x \in {{\mname{SNoS\_}}~{\beta}}\}}})}}~{\mname{Empty}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & cfe1f500f4a75d5fa0a4a8a1366f200cc39ec6e6179ec1c26eadd7b4611de086 \\
Pure Prop Address: & TMPQvhtityMjoibgERv3aQLgbEgRcqm7yXL \\
Theory Prop Id: & ee8e7c5eccf8aee691d357e43b94c01938268230735907d55299cc3a35954d02 \\
Theory Prop Address: & TMQPK7b3tUhBwCLXYG48aj1mc232WYpgPJF
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfpgchegjgogbgmgpfpgchegjgogbgmg} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad beta .  {{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{\mname{ordinal}}~{({{\alpha}+{\beta}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e9309ac99999e13ed9ba4be238444a0ade5ba236e0ed518f2e0b024d43d638d7 \\
Pure Prop Address: & TMQJ6hARogVv6a8W1vmi4QLJdstmu7vvsko \\
Theory Prop Id: & ee9c238cf553597102f9b9ee4da351edf8550e6a9298c9f211be331552a2cb2b \\
Theory Prop Address: & TMRRB6iU5Y8ULDEdxSDS6ZdHUDYdozzKnx2
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfpgchegjgogbgmgpfdfme} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad beta .  {{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{{{\mname{ordsucc}}~{\alpha}}+{\beta}}={{\mname{ordsucc}}~{({{\alpha}+{\beta}})}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d55bd8e80d14016f86e065ddbbdfb226571ce2f30e4aebe8b4e0aca9e5092472 \\
Pure Prop Address: & TMQnHkNidg5FWQLhSNenPvjDxCjnJdeF1bF \\
Theory Prop Id: & 61ae57513d7783186f7327027f1807ee424c8d2af57399e9fb52c5685603f457 \\
Theory Prop Address: & TMcBJQnghsqzp4UQfnx4PBJi2B8VvW3agRn
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfpgchegjgogbgmgpfdfcf} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad beta .  {{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{{\alpha}+{{\mname{ordsucc}}~{\beta}}}={{\mname{ordsucc}}~{({{\alpha}+{\beta}})}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e7378a22af92b7c147bbd7898baff5455a0ef96fa00ffdf898122a3bf428bdd3 \\
Pure Prop Address: & TMXcxTEQ97P4n3PTFXCCqgWyQErB5tL9Za7 \\
Theory Prop Id: & 0c1a6df8d4077eb378e14555b16968f4b38ec67e6f97bbeefe5a78c70e15a227 \\
Theory Prop Address: & TMLPo24YiNr9FHf939LUvMLAynxujJ8Qhjg
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfpgchegjgogbgmgpfjeogme} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad beta .  {{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{\unicdcdadad gamma \in {\alpha} .  {{{\gamma}+{\beta}} \in {{\alpha}+{\beta}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 09a829e8ab6f31d46768fdcb929196886d6328bcf64c285f93a561a65c1398aa \\
Pure Prop Address: & TMRVsTCaSviQYts1RQeAwzNWGiuAUb5kYUF \\
Theory Prop Id: & 5acda8957c70e073f723185142d6e08818fa09b36f05b1bd211a60f1ba14fdf9 \\
Theory Prop Address: & TMVTmvXxRgnniTCwKdyop3YHpKGaSbz6KZo
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfpgchegjgogbgmgpfjeogcf} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad beta .  {{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{\unicdcdadad gamma \in {\beta} .  {{{\alpha}+{\gamma}} \in {{\alpha}+{\beta}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d291f44187e6051da6ee1d562ded685ef47c231d1bf1f5d37d7dda55a8e5ca3f \\
Pure Prop Address: & TMNJwMYCWAjhoPBNGGAFssvdn2KaxT6X5zC \\
Theory Prop Id: & 0f42b5bcdea755e828207f260bca45c0356328488660727bd8cab53262b609b3 \\
Theory Prop Address: & TMSm47BPo5yfVb687cN4t62bbsfGbbbXCiA
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfogbgehpfbgegegpfdfoepg} ${\unicdcdadad n m \in {\mname{omega}} .  {{{{\mname{add\_nat}}~{n}}~{m}}={{n}+{m}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1701fc5ed7579abb5b201f5ec8a96acd2ac68046ad3176c036d134dea5669afe \\
Pure Prop Address: & TMXMKYNwgEJbbeSZcXdj5a87FYKoc4K9apt \\
Theory Prop Id: & 9df659a7b32d9f3afa10987ffbc9d9a84706cca922950d969a3d81f702e1b893 \\
Theory Prop Address: & TMGmoXE93EaXQanV3zMwj3Fp6HW7SWjG9xx
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfjeogpfpgngfghgbg} ${\unicdcdadad n m \in {\mname{omega}} .  {{{n}+{m}} \in {\mname{omega}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f45cbb9681681aaabac9270b640eb30171eb36f216ec7a99e5823ef45f94066a \\
Pure Prop Address: & TMPib41CNAq7UTPnewz4ZBfR7FdVnYWEtRy \\
Theory Prop Id: & eb6375250d942b2db2172dc601d939190506da07644a1cc5fdce38ab22ecf5fa \\
Theory Prop Address: & TMP489nANG759rumydRqmyjQssxqD72fosB
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdfoepgmepfjgogehfgchahpgmgbgehfg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad u \in {{\mname{SNoL}}~{({{x}+{y}})}} .  {{({\unicdcdaddd v \in {{\mname{SNoL}}~{x}} .  {{u}\unicdcdgded{{v}+{y}}}})}\unicdcdcdid{({\unicdcdaddd v \in {{\mname{SNoL}}~{y}} .  {{u}\unicdcdgded{{x}+{v}}}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 97136316ef3ba2048fc844999ceaa34b07167a3712ce76f041d8382dbfc0b380 \\
Pure Prop Address: & TMFeNW7HkxH1nH7hkD7ZZwNKgZJpFXGy2jK \\
Theory Prop Id: & 3cafc181e5e0dd3820d6f9d73c0ed6639ab60e79224ef8a2c5119732a877cd30 \\
Theory Prop Address: & TMMVRDZraKVB3LvPdW6kCBnJker3CjaSPfz
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdfoepgcfpfjgogehfgchahpgmgbgehfg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad u \in {{\mname{SNoR}}~{({{x}+{y}})}} .  {{({\unicdcdaddd v \in {{\mname{SNoR}}~{x}} .  {{{v}+{y}}\unicdcdgded{u}}})}\unicdcdcdid{({\unicdcdaddd v \in {{\mname{SNoR}}~{y}} .  {{{x}+{v}}\unicdcdgded{u}}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 928f71993ee178fd38105f25f4ad163aeeb97d5ac1bbb4c218f4c3b81d4cd25f \\
Pure Prop Address: & TMX122A6CkJ17L11g6XTAB5Z87Xwqzq3F4H \\
Theory Prop Id: & 31fa9428aeddf723102cc0d5660dce0ce0e1e068f2bd5d6fd5d853418947a80e \\
Theory Prop Address: & TMTKJ9smZabCsGAHoewvLNHykWiCKJuNchm
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfbgdhdhpgdg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}+{({{y}+{z}})}}={{({{x}+{y}})}+{z}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 39b331b6381b6e5ee82cce99630525f4cb0ca03008347a4acf301031da6002ce \\
Pure Prop Address: & TMUConoLia3kWzpgXw8RzijjEGkNtvGJecF \\
Theory Prop Id: & bc15ad70cce2a78c1f054e4e77fa522dd72c07d6620fb831284863016cf6e3da \\
Theory Prop Address: & TMFHBgSLqG7xXui9hnNbAKuSE67kqnChBbz
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdgbgogdgfgmgpfme} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{{x}+{y}}={{x}+{z}}}\unicdbdjdcd{{y}={z}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0140969cf9285d05d4b30bf21c910adfed405122d410120abd72c11f5ab69789 \\
Pure Prop Address: & TMV4roc7BLNMpAapZh7KA7NtFj11mPqdozE \\
Theory Prop Id: & 781256c452aceda44607ee4205e8ee86d1eea0b3f32743f53b342771142ce743 \\
Theory Prop Address: & TMHbLqftV2LcfDUv7HqxqMhm7CVsY7Ef2K8
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdgbgogdgfgmgpfcf} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{{x}+{y}}={{z}+{y}}}\unicdbdjdcd{{x}={z}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3de43c040780ea5cd150c912ec06fac7c4f463b8b0a89ff1d2cba1c673d3b3d6 \\
Pure Prop Address: & TMLNfa9NDWM7B9HYGBe7xuczocSytHXKARC \\
Theory Prop Id: & e62801806a6a72a797723a605192e13732faf66171abd8d9323db7972673a977 \\
Theory Prop Address: & TMVQQYhZPbjCpQtrWniV5GvvRBpj4GaPPzW
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfbgegegpfdfoepgpfegjgdhehch} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{\prenc{{({{x}+{y}})}}}={{({\prenc{{x}}})}+{({\prenc{{y}}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 57ec9da45362cd7c9418befe3fb2ca772e55318bd76cfac89139baf603dd198c \\
Pure Prop Address: & TMQTvHftZnQzgoUvhX3DhSiB9EYGcjbMEVt \\
Theory Prop Id: & da8d5a3c4332f78b540e7fb374f3ba393f34ebef39aa08f03a43e8a0eb0ec48f \\
Theory Prop Address: & TMMrTspPpgWYYtPV5GfMyFKZxbTpr6YpBF8
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfbgegegpfdfoepgpfegjgdhehchpfdd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{\prenc{{({{x}+{{y}+{z}}})}}}={{\prenc{{x}}}+{{\prenc{{y}}}+{\prenc{{z}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b074acb051db50b8aebd6c84b804f36ef95546c8c1d8b4a12fad41b6036b675d \\
Pure Prop Address: & TMHG5Wzxu3ufeWeCBHAySJy5K1FsFRffozM \\
Theory Prop Id: & 94dbf3383e5ad97c01e0494f7ad58f05eb86370106ca98f801505d5cc17d8e8c \\
Theory Prop Address: & TMGDtLEgxm1wM1omKuqmT7qds1DiL446e61
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmefgghpfcgeg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNoLev}}~{({{x}+{y}})}} \subseteq {{{\mname{SNoLev}}~{x}}+{{\mname{SNoLev}}~{y}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0f257c03f4bd792cfafdf5b6628524966280c12bc90c0b7152a74c1c453b0a97 \\
Pure Prop Address: & TMHLNim923vpzTQhz7fmKu2bm4ncVWUmWEE \\
Theory Prop Id: & 0680effc26564a3378ad068703ae226003b02850f73e91331345c41424701bcf \\
Theory Prop Address: & TMKzYXGUEBsqkz6Pd1y7TGUmq9kreetmiSK
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdfoepgdfpfpgngfghgbg} ${\unicdcdadad x y \in {{\mname{SNoS\_}}~{\mname{omega}}} .  {{{x}+{y}} \in {{\mname{SNoS\_}}~{\mname{omega}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9311032f024de4324b4cbbbc04cbbc6f507e4664b92e5daf8d2b539deaa0d877 \\
Pure Prop Address: & TMRdieTgkX8dUfZyNNYHypf4kTH3PoJW3Ms \\
Theory Prop Id: & 6d16aa7a6b095f74f1b34091c0e9a002aa247761437c60a7c0261e4d1687a1d4 \\
Theory Prop Address: & TMVjF1tKn3421u44CyHtBcSNvHoJBqRKST1
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfcfcd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{({{x}+{y}})}+{\prenc{{y}}}}={x}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3e7b333f3cd5354e81de9e18e82adb1ea3101121fa6de8717d4d9dd7a8de2dfc \\
Pure Prop Address: & TMbjFdWsQkwJkFdFKsygecXhLiJUroenET2 \\
Theory Prop Id: & 91f4efb884c7f253c44c5fd69b6ef119bd466cbf203a587487e6a292ebc37f9e \\
Theory Prop Address: & TMdam15xQW7pR7unsMT5iH6SPdFATTi712m
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehbdpfdgbgogdgfgmg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{{x}+{y}}<{{z}+{y}}}\unicdbdjdcd{{x}<{z}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e5c11e1cc1b83566aacf4247c67eda38cbd881bcf48b4a30fd81d1ae9be3ae25 \\
Pure Prop Address: & TMVGk6XaUoPX6ZGVPLEVZWzvzKkLektMYLP \\
Theory Prop Id: & 2ec8aef4360c9ee2ffa1912943451de1852b1cf2af0400f6da375d69c467ae9e \\
Theory Prop Address: & TMLJkdMCCCn6YVEdM7uNU6PFwazJo7w6REd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehcdpfdgbgogdgfgmg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{{x}+{y}}<{{x}+{z}}}\unicdbdjdcd{{y}<{z}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9ecfc82c2e1b3fbbf14127fd056f50b5f4f4058c7b9c2da324e8e11162871620 \\
Pure Prop Address: & TMJyFpf59yM4gCj8nhUhoxe5MLMc9ZvK1hj \\
Theory Prop Id: & d4afbdcdb563fce45f9fdb7e6d26326d06598fc21bee88b552fb7439b946957b \\
Theory Prop Address: & TMF8QKcffVRxMAdSUMqMY8FSyHY2Pu3AYJc
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfbgdhdhpgdgpfed} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{x}+{{y}+{{z}+{w}}}}={{({{x}+{{y}+{z}}})}+{w}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b9919a859ce250ebed40a38e210d51ea56ad81bd3b548d2973c10fa4b77224fb \\
Pure Prop Address: & TMNn2vs2scHFb5sPooErQtoePq5skwF4Rqs \\
Theory Prop Id: & f2ec8cf0669d6b7f4ba80ac3f1c7fc3a38bc969d82b8ab446307e654aeb60211 \\
Theory Prop Address: & TMJdJZYxC1SYeUZNH9CnoUzpc5nbzLd4E1K
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdgpgngpfddpfadpfbd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}+{{y}+{z}}}={{y}+{{x}+{z}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 53adb5589660e895a568e553cecd4c833dfe11b8d4b0a59395733cf1473d5a4f \\
Pure Prop Address: & TMan13aAzSS4boU3k6BCV2m9HpcAqzjMqYs \\
Theory Prop Id: & 3c9cc8b1863d8757dc461ba3222ea23daf40651a26eb8bbaa4eb74d8ce5ecb34 \\
Theory Prop Address: & TMLUi74QYWaLowsgzo8D5KaFFmMGAF33Uov
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdgpgngpfedpfjgogogfgchpfggmgbgeh} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{x}+{{y}+{{z}+{w}}}}={{x}+{{z}+{{y}+{w}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 353c422c4473fec6d253f64dc6011273518f6fd9b8666b1d04ac4b7a10b21676 \\
Pure Prop Address: & TMYagwq4AL9eS2satFuJrRGsXJQ7UaQ2Y2Z \\
Theory Prop Id: & d32d7fe4fd859c93a5b059b561d61395eae0d2dc1771536899a51aa7290836c3 \\
Theory Prop Address: & TMJ7ALWqQknTtDK5LC67JaovH5kH3hMjM3S
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdgpgngpfddcgpfbdpfcd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{({{x}+{y}})}+{z}}={{({{x}+{z}})}+{y}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2ccf2bcc11ca672b64552dc368e1746da109c4c422678991b3ab7c3011849772 \\
Pure Prop Address: & TMPuoTqK9f79je6oZPXi6p9wEreuXeYKFpg \\
Theory Prop Id: & f2dd80a1f9d4848b30eb661596f9b7355f21ffc0ff2b67d04df7791c06f27387 \\
Theory Prop Address: & TMXQdGRpcSTBjZ8pjJmoA2DCTzkK5tBAKzR
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfdgpgngpfedpfjgogogfgchpfngjgeg} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{({{x}+{y}})}+{({{z}+{w}})}}={{({{x}+{z}})}+{({{y}+{w}})}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 635cb763d6883b1df327b97d296852bd72be35ebf27ba3197cc20d58b5db1923 \\
Pure Prop Address: & TMNBs9W7CDbVpEM2DJAD6LJaUJdzLs5mL2p \\
Theory Prop Id: & e25ee6de8e445a677f9f91027aeb3dc6ab5acb07b724880668b83cb66a9d287c \\
Theory Prop Address: & TMRqXvHGUrVc7Tj4T8TaayZuLEe7vs6QkAa
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfchpgehbgehfgpfddpfbd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}+{{y}+{z}}}={{z}+{{x}+{y}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 51654e504536bbcc3c5f757f3dcad5e306f61cb3f3113fe097e1d1fbba5aac97 \\
Pure Prop Address: & TMNHRTRCD9bsMPb3UbC8jvRrJ6N9ZH1Eh4N \\
Theory Prop Id: & 340640be61f33faa3a7f153679d19bb2ce9291b5efec590860ba767ade73a9a3 \\
Theory Prop Address: & TMQNLhVAcRPvXzedEwagyyteqQKc581dhD5
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfchpgehbgehfgpfedpfbd} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{x}+{{y}+{{z}+{w}}}}={{w}+{{x}+{{y}+{z}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4bc7aa1aee0011ef665476863c178f590fbe4fbe6699dd7ccb934845846d8dd3 \\
Pure Prop Address: & TMaTstSL13vyKcGtLbTNn8eAVSGWCpDMRRP \\
Theory Prop Id: & ce0a6b44cbcebe4b78d9bcbc4db2e540564f76969de5d18f22a263048dfe8518 \\
Theory Prop Address: & TMYrz7XuuFsQmh65FTDfV3W2dcW9j9WhNCe
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfchpgehbgehfgpffdpfbd} ${\unicdcdadad x y z w v .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{\mname{SNo}}~{v}}\unicdbdjdcd{{{x}+{{y}+{{z}+{{w}+{v}}}}}={{v}+{{x}+{{y}+{{z}+{w}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4b4b645a9de69d196bcfac0f564a76aba4c24d5518f13324b64621276d543128 \\
Pure Prop Address: & TMSSCZTFkPsvFM1mXZ7MP4aihGyFoBRfhdC \\
Theory Prop Id: & 0714aabd91b09f4e30b2978cd4c05acb5975913fcf91782f98282a06ecc98e76 \\
Theory Prop Address: & TMKsqf29EQpew5TASzktuUb1DHqMTYo9542
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfchpgehbgehfgpffdpfcd} ${\unicdcdadad x y z w v .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{\mname{SNo}}~{v}}\unicdbdjdcd{{{x}+{{y}+{{z}+{{w}+{v}}}}}={{w}+{{v}+{{x}+{{y}+{z}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 33b5cc61253fa4c067f352dd7d4f385b800a18a8d891459904b242726be1c98b \\
Pure Prop Address: & TMNP66fS26FKAKuh9MEtRW9VDian3FYj9qr \\
Theory Prop Id: & fec9d019d82128a582a52e0bc0e64cfc44a05d4f84f9952d8cb92946b031856d \\
Theory Prop Address: & TMUguce5TL82mq83tgNMXbiPmFe3zcgvQFJ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfdfoepgpfahchpgahbd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\prenc{{x}}}+{{x}+{y}}}={y}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2cd65e577fae467be069f5ab9a2bfa88c578085e70deca674f146625df01bb31 \\
Pure Prop Address: & TMNj1ZknUmnj8fyskPShRtyeBwzp2553NVN \\
Theory Prop Id: & 598ee6b17ec7a3c47d15dffa959d384acc4d4d5bdbcd02790c13e2ff045e1440 \\
Theory Prop Address: & TMKE2xaN2B6wD8vGVfwUJaufhki7EhAkGfE
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfdfoepgpfahchpgahcd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}+{{\prenc{{x}}}+{y}}}={y}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 632a4d661bba9435289e33a84ddfd6e1ebe55256f17529d37fd7bf784a245260 \\
Pure Prop Address: & TMFUMMG5NWfvwVqV1VzWhtid5D1cW3ENDqd \\
Theory Prop Id: & a8450da6ba6ffd91e010f90197162261ef21fb4e025e591e5137531e12d7ad70 \\
Theory Prop Address: & TMJQtEivAHnWG6vm64j8jL7rdfeCLJv59AH
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfdfoepgpfahchpgahdd} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{({{x}+{{y}+{z}}})}+{({{\prenc{{z}}}+{w}})}}={{x}+{{y}+{w}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 038fdd7f7b41d5dbabb1c2b43f6ee18b5eb51c7b73b574795aa81a5f6e7185db \\
Pure Prop Address: & TMQQTF6qMromiAnZNHGPbLUmDnoRJJsw184 \\
Theory Prop Id: & 0d4f2459906cf0deac11ccfa5d8ce0300b8a0e2e2e35908d116f376387aeaf89 \\
Theory Prop Address: & TMVtSJwaz4PKT2M9971YpjjmSqY2uXJQFDe
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfdfoepgpfahchpgahed} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{({{x}+{{y}+{z}}})}+{({{w}+{\prenc{{z}}}})}}={{x}+{{y}+{w}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9303242681e65d4bb41186882b5827d87cfb757866a860dcd21e8d85c6540d00 \\
Pure Prop Address: & TMZxz5UVrwqNzRN8eXdiojfqVbJhkXRzfpt \\
Theory Prop Id: & 4c3067acc33707997f49c35d6a0ac8a1b7845a5ed0454d39acbfe19ba57d0c1f \\
Theory Prop Address: & TMWftxv2baJqYmzmKYPcnYgd25XrtWoXJST
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfdfoepgpfahchpgahfd} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{({{x}+{{y}+{\prenc{{z}}}}})}+{({{z}+{w}})}}={{x}+{{y}+{w}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & bdfd3ff33cf2f7c3beec961e55cb8c70e3e72bedc8e24a5079ae13e57cef924e \\
Pure Prop Address: & TMbd9Mg6tPihAjiCRTdu2H9iHTJu8P6vdH9 \\
Theory Prop Id: & eeb587de80f26e1b6307d8991dce2c6ed3d960c0a1571b8722cff655214df698 \\
Theory Prop Address: & TMRmP8fuJtFdvTRXG6bJ2zsSkasVFLW8YiM
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfmeehbd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{{x}+{\prenc{{y}}}}<{z}}\unicdbdjdcd{{x}<{{z}+{y}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 643876090ca9e8c818bb1d1517026337499a8a61a6383863a9f6b90713b953f8 \\
Pure Prop Address: & TMHorPzXGJj6NLSDuVdotitbnKv8te63hAg \\
Theory Prop Id: & 4033dc222689776035a38336a7f6ce157593e8786381729a207ecdc69f8508d1 \\
Theory Prop Address: & TMGgmNN1c8BXxQunMKeC7ctpH6iBXxvzUXy
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfmeehbdcg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}<{{z}+{y}}}\unicdbdjdcd{{{x}+{\prenc{{y}}}}<{z}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5012b14b32060175e560f0d1e7e017da053c610000fa170aeaf365e6d6410d7c \\
Pure Prop Address: & TMGs3ZQCrCC6dtPSMkxhL3RGUTqUer9SdGs \\
Theory Prop Id: & 0b5cca84918d73e36a13fd959569d5c6a5d4a05c85317235433f59a19bc601a2 \\
Theory Prop Address: & TMPp2LmW4bmLHnLXjmQuZtQpoHAz4uTwF1i
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfmeehcd} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{z}<{{x}+{\prenc{{y}}}}}\unicdbdjdcd{{{z}+{y}}<{x}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 76c543653b6da539ad0d8357582da1ab031b9383d2fae35a230e98de8946856e \\
Pure Prop Address: & TMFL2J2Xi9knH1EYexCMF24iaPSDRahCQtP \\
Theory Prop Id: & 3621456ff64800b7e93dff3968e21ba0755569c11a30efbf0360787c625c56dc \\
Theory Prop Address: & TMSsbaxypo5H5nF8jvxTgrgpmxmVfGPHE3p
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfmeehcdcg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{{z}+{y}}<{x}}\unicdbdjdcd{{z}<{{x}+{\prenc{{y}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 24d1a6c7194f01d2e61624e193908bc5ed6870b188d549fdbfafb81f3e69660b \\
Pure Prop Address: & TMYLxBcGLaYBenPqsTbU48VLhTDu35ZvWin \\
Theory Prop Id: & 68419210383aa804b0c97019a1190bf0a75fb45009b55d3f310cccb98a55f216 \\
Theory Prop Address: & TMLRXmb9LNiNLw5Y1urKn6AQHWWwgKREUkY
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfmeehbdcgdd} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{{x}+{y}}<{{w}+{z}}}\unicdbdjdcd{{{x}+{{y}+{\prenc{{z}}}}}<{w}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 218e4f25517f8ff1478046e3df782ae832d9f7e33e0b433bf05b9643a0208e85 \\
Pure Prop Address: & TMRSZGz7nE2AVHGmgMaepm9CAmfWzJ72Boq \\
Theory Prop Id: & 8ea4a7cab08aeec1b407bd563474dd6766fbcf0794f34b125012f4f8d61f021f \\
Theory Prop Address: & TMRrXMsdvggvcKCwoxKF8kLUbznUq5NaVmd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfngjgogfhdhpfmeehcdcgdd} ${\unicdcdadad x y z w .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{{w}+{z}}<{{x}+{y}}}\unicdbdjdcd{{w}<{{x}+{{y}+{\prenc{{z}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d9462b9396765bcb6e599c448b5b2a0e15bfeea62de5713637e71fe91bc14e06 \\
Pure Prop Address: & TMG6X9g8zK9X5PbsVyTtDpJgN73SyRkZBJ2 \\
Theory Prop Id: & 4bf57517c0115e8dae375c95d1c1bf403694f4ee328c9dfd9c34f22eca40d62b \\
Theory Prop Address: & TMGSKHUM3KGisvSLGi1Zpy9MG4UtFxqFThU
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehpfdhfhcgahchpgahcd} ${\unicdcdadad x y z w u v .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{\mname{SNo}}~{u}}\unicdbdjdcd{{{\mname{SNo}}~{v}}\unicdbdjdcd{{{{x}+{u}}<{{z}+{v}}}\unicdbdjdcd{{{{y}+{v}}<{{w}+{u}}}\unicdbdjdcd{{{x}+{y}}<{{z}+{w}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 848909edfa645bd860cf5cc6acec778e23e88c5a769fe3c959d4b3566fbb306b \\
Pure Prop Address: & TMLUrjDQwdJ3cSJVaNX11wZPbmHmYJm2F4A \\
Theory Prop Id: & 7f1b50ad173ad970e6eee06a0d1dedb182caf1cfff89406b30003a7afc38c6c3 \\
Theory Prop Address: & TMcJPm4yGSHS1kUSPRFgEHGenN7vG4V1Nzw
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehpfdhfhcgahchpgahddbg} ${\unicdcdadad x y z w u a .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{\mname{SNo}}~{u}}\unicdbdjdcd{{{\mname{SNo}}~{a}}\unicdbdjdcd{{{{x}+{z}}<{{w}+{a}}}\unicdbdjdcd{{{{y}+{a}}<{u}}\unicdbdjdcd{{{x}+{{y}+{z}}}<{{w}+{u}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 43322d509ad5f4223a3ea2d226e4b0ccb8c26b9a1175adae9d1d447e776bf7c0 \\
Pure Prop Address: & TMMYB2kCLPC8xoSmU63JvGfNSPFzUYuK6pv \\
Theory Prop Id: & cde0c9b5a2910f46768762e1f3849c4e1b8cda57c9c1575aa1b9bf556baa627d \\
Theory Prop Address: & TMPNtwtWveLBYLorWYj4fRoEpqz56R9oTm5
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehpfdhfhcgahchpgahddcg} ${\unicdcdadad x y w u v a .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{\mname{SNo}}~{u}}\unicdbdjdcd{{{\mname{SNo}}~{v}}\unicdbdjdcd{{{\mname{SNo}}~{a}}\unicdbdjdcd{{{{x}+{a}}<{{w}+{v}}}\unicdbdjdcd{{{y}<{{a}+{u}}}\unicdbdjdcd{{{x}+{y}}<{{w}+{{u}+{v}}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9d5c37e2e6a26c6dc29e00cf8767d02fc77b30d09cebed0550f159cb9d1334c6 \\
Pure Prop Address: & TMakKb23RRtRLWKpGuoMNwnASjKertEftag \\
Theory Prop Id: & b197b177985ff32095d16adfad88ff6b97c3c9885a5a1cadc19c5cb553b4345d \\
Theory Prop Address: & TMduddfsYHhZALrRDzTwzypdKDVHK1k8VYh
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehpfdhfhcgahchpgahdddg} ${\unicdcdadad x y z w u a b c .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{\mname{SNo}}~{u}}\unicdbdjdcd{{{\mname{SNo}}~{a}}\unicdbdjdcd{{{\mname{SNo}}~{b}}\unicdbdjdcd{{{\mname{SNo}}~{c}}\unicdbdjdcd{{{{x}+{a}}<{{b}+{c}}}\unicdbdjdcd{{{{y}+{c}}<{u}}\unicdbdjdcd{{{{b}+{z}}<{{w}+{a}}}\unicdbdjdcd{{{x}+{{y}+{z}}}<{{w}+{u}}}}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 18461471e206fc8672b0e953edf289af2051148c543ab9c39752e75fd74b9902 \\
Pure Prop Address: & TMLDVH1R1XQXWmhFyN7kxcnxZ8vV1jj1gge \\
Theory Prop Id: & f8cf2ba034b2ae66a187647fe084618a9c62057a7ab092702eb1355e51bfdb56 \\
Theory Prop Address: & TMNUyCkvtK5vp56fDjYpDWz4tisomP2mVS5
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfmeehpfdhfhcgahchpgahddeg} ${\unicdcdadad x y w u v a b c .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{w}}\unicdbdjdcd{{{\mname{SNo}}~{u}}\unicdbdjdcd{{{\mname{SNo}}~{v}}\unicdbdjdcd{{{\mname{SNo}}~{a}}\unicdbdjdcd{{{\mname{SNo}}~{b}}\unicdbdjdcd{{{\mname{SNo}}~{c}}\unicdbdjdcd{{{{x}+{a}}<{{b}+{v}}}\unicdbdjdcd{{{y}<{{c}+{u}}}\unicdbdjdcd{{{{b}+{c}}<{{w}+{a}}}\unicdbdjdcd{{{x}+{y}}<{{w}+{{u}+{v}}}}}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a447c6d932bea68dbde54b06fe58a08cf250a32d1e9e252f9aa646d7d450a31f \\
Pure Prop Address: & TMGmodxxtB7938aypz74aGsuhYxLsXh57g5 \\
Theory Prop Id: & a1520a522ad3deb92a6746488e5cfc1dfb1d1fd6684fe79e87bba20cda4756b6 \\
Theory Prop Address: & TMLrgfSnAmTdRkGo9xWrngAvimcZVK9yU2d
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:pgchegjgogbgmgpfpgchegdhfhdgdgpfdfoepgpffgbh} ${\unicdcdadad alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{ordsucc}}~{\alpha}}={{1}+{\alpha}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f0596d71a0270ee871aebb4abb6a468185b1e2d2deec29fda29af07ac94080e7 \\
Pure Prop Address: & TMKvpBUtRCutc7AQ2jjzLG87zkaCypGHwqw \\
Theory Prop Id: & 36bf6dd6ade137d7a6431b1d2f74c22803884fe3bcc9f6b223f90c9587bc1fde \\
Theory Prop Address: & TMM3U1W3pWWJb6gkvcZQNx2QDYDK4UxBhJx
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfpgngfghgbgpffgahdhpfmeeh} ${\unicdcdadad x y \in {{\mname{SNoS\_}}~{\mname{omega}}} .  {{{x}<{y}}\unicdbdjdcd{\unicdcdaddd n \in {\mname{omega}} .  {{{x}+{{\mname{eps\_}}~{n}}}<{y}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5a850b3a5f7e8e409aa7b3a8fde3eaf603b6c57ac0565694a8d02fbe81aa1496 \\
Pure Prop Address: & TMSCHeh28B5U2XMhbESr5SKSXVCskfzuwZx \\
Theory Prop Id: & 0afdf612e70b6faf98df67163f00af4cacf8720792da2450f3f84ab2b65e27b4 \\
Theory Prop Address: & TMYHZueBTibUJWtZVQUjiVzfGCegt89U2VT
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:ngfhmgpfdfoepg}
$\mname{mul\_SNo}$ is the opaque object of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 48d05483e628cb37379dd5d279684d471d85c642fe63533c3ad520b84b18df9d \\
Pure Object Address: & TMSw7TH4yUrNvqkxFpRdKgaGjhbMXKDtVKM \\
Theory Object Id: & b14730ddd09967223586b7b73ba059d8fee46ef74b4d5c13b7276c084e8f3dea \\
Theory Object Address: & TMdQsgnzh6YiiBs9ki4XhdobvDFEYV9QYKG
\end{tabular}
\end{center}

\end{definition}
{\bf{Notation.}} We use $\infkc$ as a right associative infix operator  corresponding to applying term ${\mname{mul\_SNo}}$.
\begin{theorem}\label{thm:ngfhmgpfdfoepgpffgbh} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{\unicdcdadad y .  {{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}*{y}}={{{\mname{SNoCut}}~{({{\{{{{({{w}~{0}})}*{y}}+{{{x}*{({{w}~{1}})}}+{\prenc{{{({{w}~{0}})}*{({{w}~{1}})}}}}}}|w \in {{{\mname{SNoL}}~{x}}\unicdbgcdgg{{\mname{SNoL}}~{y}}}\}}\unicdcdcdbg{\{{{{({{z}~{0}})}*{y}}+{{{x}*{({{z}~{1}})}}+{\prenc{{{({{z}~{0}})}*{({{z}~{1}})}}}}}}|z \in {{{\mname{SNoR}}~{x}}\unicdbgcdgg{{\mname{SNoR}}~{y}}}\}}})}}~{({{\{{{{({{w}~{0}})}*{y}}+{{{x}*{({{w}~{1}})}}+{\prenc{{{({{w}~{0}})}*{({{w}~{1}})}}}}}}|w \in {{{\mname{SNoL}}~{x}}\unicdbgcdgg{{\mname{SNoR}}~{y}}}\}}\unicdcdcdbg{\{{{{({{z}~{0}})}*{y}}+{{{x}*{({{z}~{1}})}}+{\prenc{{{({{z}~{0}})}*{({{z}~{1}})}}}}}}|z \in {{{\mname{SNoR}}~{x}}\unicdbgcdgg{{\mname{SNoL}}~{y}}}\}}})}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 726bd964aa39e59e3a6693e9b7d4306b1accb9a6c62ae1891b050b131c4ea01f \\
Pure Prop Address: & TMSnEDskXoZEWD5U2DVgxtTRUp75JLmq8fL \\
Theory Prop Id: & 3525c5e458ce4f160f6f38c46b032eb43534d43481b8d0ff0b1096daa93e9080 \\
Theory Prop Address: & TMR2SBeeUCWzXPrVEFQprrZoRPjqi24Acwd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpffgbhpfcd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad p : {o} .  {{({\unicdcdadad L R .  {{({\unicdcdadad u .  {{{u} \in {L}}\unicdbdjdcd{({\unicdcdadad q : {o} .  {{({\unicdcdadad w0 \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad w1 \in {{\mname{SNoL}}~{y}} .  {{{u}={{{w0}*{y}}+{{{x}*{w1}}+{\prenc{{{w0}*{w1}}}}}}}\unicdbdjdcd{q}}}})}\unicdbdjdcd{{({\unicdcdadad z0 \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad z1 \in {{\mname{SNoR}}~{y}} .  {{{u}={{{z0}*{y}}+{{{x}*{z1}}+{\prenc{{{z0}*{z1}}}}}}}\unicdbdjdcd{q}}}})}\unicdbdjdcd{q}}}})}}})}\unicdbdjdcd{{({\unicdcdadad w0 \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad w1 \in {{\mname{SNoL}}~{y}} .  {{{{w0}*{y}}+{{{x}*{w1}}+{\prenc{{{w0}*{w1}}}}}} \in {L}}}})}\unicdbdjdcd{{({\unicdcdadad z0 \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad z1 \in {{\mname{SNoR}}~{y}} .  {{{{z0}*{y}}+{{{x}*{z1}}+{\prenc{{{z0}*{z1}}}}}} \in {L}}}})}\unicdbdjdcd{{({\unicdcdadad u .  {{{u} \in {R}}\unicdbdjdcd{({\unicdcdadad q : {o} .  {{({\unicdcdadad w0 \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad z1 \in {{\mname{SNoR}}~{y}} .  {{{u}={{{w0}*{y}}+{{{x}*{z1}}+{\prenc{{{w0}*{z1}}}}}}}\unicdbdjdcd{q}}}})}\unicdbdjdcd{{({\unicdcdadad z0 \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad w1 \in {{\mname{SNoL}}~{y}} .  {{{u}={{{z0}*{y}}+{{{x}*{w1}}+{\prenc{{{z0}*{w1}}}}}}}\unicdbdjdcd{q}}}})}\unicdbdjdcd{q}}}})}}})}\unicdbdjdcd{{({\unicdcdadad w0 \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad z1 \in {{\mname{SNoR}}~{y}} .  {{{{w0}*{y}}+{{{x}*{z1}}+{\prenc{{{w0}*{z1}}}}}} \in {R}}}})}\unicdbdjdcd{{({\unicdcdadad z0 \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad w1 \in {{\mname{SNoL}}~{y}} .  {{{{z0}*{y}}+{{{x}*{w1}}+{\prenc{{{z0}*{w1}}}}}} \in {R}}}})}\unicdbdjdcd{{{{x}*{y}}={{{\mname{SNoCut}}~{L}}~{R}}}\unicdbdjdcd{p}}}}}}}}})}\unicdbdjdcd{p}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 97f72a94a57b9c976cabb7db6e1c9e782baccf6d99699b64f549f38a4a8cb50c \\
Pure Prop Address: & TMLuSvvCfD9Ze7AKPowkWZPkeM2o9mvreGc \\
Theory Prop Id: & d69cc6521bcd2b18a3af424dfe57b3ca7eccc3c292bd43c9f65e2a98dcebcffe \\
Theory Prop Address: & TMRezz7usQBo1aanPbksRyAGNHbrpgWVes4
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfahchpgahpfbd} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{\unicdcdadad y .  {{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad p : {o} .  {{({{{\mname{SNo}}~{({{x}*{y}})}}\unicdbdjdcd{{({\unicdcdadad u \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad v \in {{\mname{SNoL}}~{y}} .  {{{{u}*{y}}+{{x}*{v}}}<{{{x}*{y}}+{{u}*{v}}}}}})}\unicdbdjdcd{{({\unicdcdadad u \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad v \in {{\mname{SNoR}}~{y}} .  {{{{u}*{y}}+{{x}*{v}}}<{{{x}*{y}}+{{u}*{v}}}}}})}\unicdbdjdcd{{({\unicdcdadad u \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad v \in {{\mname{SNoR}}~{y}} .  {{{{x}*{y}}+{{u}*{v}}}<{{{u}*{y}}+{{x}*{v}}}}}})}\unicdbdjdcd{{({\unicdcdadad u \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad v \in {{\mname{SNoL}}~{y}} .  {{{{x}*{y}}+{{u}*{v}}}<{{{u}*{y}}+{{x}*{v}}}}}})}\unicdbdjdcd{p}}}}}})}\unicdbdjdcd{p}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3018e48e1b0b8923d2bc9e30faf03586dced5941b18f9a31b0e91b0d91716d40 \\
Pure Prop Address: & TMdaGCethjpbbQ5f2zXiMzy1Jw6trWXkwEA \\
Theory Prop Id: & 5e4d160028e2e0d69e57789a6c38f1230eca4d80e5f1e171fc8969764d28dd07 \\
Theory Prop Address: & TMZ8JpWhmNRNmu7xFhYLa8nAtcJESfVjnAP
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfngfhmgpfdfoepg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{\mname{SNo}}~{({{x}*{y}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7fa2e2dd6764040b3f888d899581ea742d829880456c04996546837d770370c5 \\
Pure Prop Address: & TMX6qVheSX36EwKWKF211NtEBN8GMqFFQjT \\
Theory Prop Id: & 9fd082b661154955f669084ede91b91cef6249e30cbec7c7a954ac883b5ca01b \\
Theory Prop Address: & TMTR3KoWZkq4VBDJf71KPEZUEwdxEBQ1SwZ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpffgbhpfdd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad p : {o} .  {{({\unicdcdadad L R .  {{{{\mname{SNoCutP}}~{L}}~{R}}\unicdbdjdcd{{({\unicdcdadad u .  {{{u} \in {L}}\unicdbdjdcd{({\unicdcdadad q : {o} .  {{({\unicdcdadad w0 \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad w1 \in {{\mname{SNoL}}~{y}} .  {{{u}={{{w0}*{y}}+{{{x}*{w1}}+{\prenc{{{w0}*{w1}}}}}}}\unicdbdjdcd{q}}}})}\unicdbdjdcd{{({\unicdcdadad z0 \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad z1 \in {{\mname{SNoR}}~{y}} .  {{{u}={{{z0}*{y}}+{{{x}*{z1}}+{\prenc{{{z0}*{z1}}}}}}}\unicdbdjdcd{q}}}})}\unicdbdjdcd{q}}}})}}})}\unicdbdjdcd{{({\unicdcdadad w0 \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad w1 \in {{\mname{SNoL}}~{y}} .  {{{{w0}*{y}}+{{{x}*{w1}}+{\prenc{{{w0}*{w1}}}}}} \in {L}}}})}\unicdbdjdcd{{({\unicdcdadad z0 \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad z1 \in {{\mname{SNoR}}~{y}} .  {{{{z0}*{y}}+{{{x}*{z1}}+{\prenc{{{z0}*{z1}}}}}} \in {L}}}})}\unicdbdjdcd{{({\unicdcdadad u .  {{{u} \in {R}}\unicdbdjdcd{({\unicdcdadad q : {o} .  {{({\unicdcdadad w0 \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad z1 \in {{\mname{SNoR}}~{y}} .  {{{u}={{{w0}*{y}}+{{{x}*{z1}}+{\prenc{{{w0}*{z1}}}}}}}\unicdbdjdcd{q}}}})}\unicdbdjdcd{{({\unicdcdadad z0 \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad w1 \in {{\mname{SNoL}}~{y}} .  {{{u}={{{z0}*{y}}+{{{x}*{w1}}+{\prenc{{{z0}*{w1}}}}}}}\unicdbdjdcd{q}}}})}\unicdbdjdcd{q}}}})}}})}\unicdbdjdcd{{({\unicdcdadad w0 \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad z1 \in {{\mname{SNoR}}~{y}} .  {{{{w0}*{y}}+{{{x}*{z1}}+{\prenc{{{w0}*{z1}}}}}} \in {R}}}})}\unicdbdjdcd{{({\unicdcdadad z0 \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad w1 \in {{\mname{SNoL}}~{y}} .  {{{{z0}*{y}}+{{{x}*{w1}}+{\prenc{{{z0}*{w1}}}}}} \in {R}}}})}\unicdbdjdcd{{{{x}*{y}}={{{\mname{SNoCut}}~{L}}~{R}}}\unicdbdjdcd{p}}}}}}}}}})}\unicdbdjdcd{p}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 51793959fac9070d8e2d9b035efecb4ad9c7ae3ca6da33d55ab3bf797bef3297 \\
Pure Prop Address: & TMaj5xJYDn5QmHnKGHqSMLP3XxL1wkJv3vo \\
Theory Prop Id: & 720ba07dcca3e0725279be9e2c27ff2f73fa21fb411e18239df4e0915a5111b7 \\
Theory Prop Address: & TMWEwehT67Ew9qx9JCyHJdWR4Zn2VcBGx6y
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfmeeh} ${\unicdcdadad x y u v .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{u}}\unicdbdjdcd{{{\mname{SNo}}~{v}}\unicdbdjdcd{{{u}<{x}}\unicdbdjdcd{{{v}<{y}}\unicdbdjdcd{{{{u}*{y}}+{{x}*{v}}}<{{{x}*{y}}+{{u}*{v}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 428b35f3bc30112249eab05959f8ad7ada3ebbb7987ced62d2fe3afde9af092b \\
Pure Prop Address: & TMLpp8YEH3knCvvN7eHZnAuNDCMtsQzd14d \\
Theory Prop Id: & c232011455c468168fd8e65dfaebb59e340331574a0089936e166d3a4fcf2f5d \\
Theory Prop Address: & TMWuSHJpH8gNAX4DUjm9Wzk6tp7vPQphtfr
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfmefg} ${\unicdcdadad x y u v .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{u}}\unicdbdjdcd{{{\mname{SNo}}~{v}}\unicdbdjdcd{{{u}\unicdcdgded{x}}\unicdbdjdcd{{{v}\unicdcdgded{y}}\unicdbdjdcd{{{{u}*{y}}+{{x}*{v}}}\unicdcdgded{{{x}*{y}}+{{u}*{v}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d9a12c5ce8058d66cd1876f20d85be2a5ceec00d4bf569f3282bbee6cb91faab \\
Pure Prop Address: & TMXSQ1sbRtw5pLGrB7wQb2i4sBb3bPWrE3x \\
Theory Prop Id: & acb62030c6b82e1dd945365185266c95047e40137c030a2512820a86905108f6 \\
Theory Prop Address: & TMFCxz59dZUG7KrmMeKtLnrjcymT35p7rac
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfdfoepgmepfjgogehfgchahpgmgbgehfg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad u \in {{\mname{SNoL}}~{({{x}*{y}})}} .  {{({\unicdcdaddd v \in {{\mname{SNoL}}~{x}} .  {\unicdcdaddd w \in {{\mname{SNoL}}~{y}} .  {{{u}+{{v}*{w}}}\unicdcdgded{{{v}*{y}}+{{x}*{w}}}}}})}\unicdcdcdid{({\unicdcdaddd v \in {{\mname{SNoR}}~{x}} .  {\unicdcdaddd w \in {{\mname{SNoR}}~{y}} .  {{{u}+{{v}*{w}}}\unicdcdgded{{{v}*{y}}+{{x}*{w}}}}}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7d1b52199f12ab9873941a7285388e03c30e7d2abddf759a627f6f95d4d06ea0 \\
Pure Prop Address: & TMd7FoqCiPFwP8mZudWTt9WbKfWC2KEFNpJ \\
Theory Prop Id: & f460fc19e5496a545e8abbaf4188a0ddb17c95a931bb76236bbcc1ab7829c3da \\
Theory Prop Address: & TMQhwb7bstgkPJXZU2FBTPqwJhZ6onk8qEw
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfdfoepgmepfjgogehfgchahpgmgbgehfgpfjgngahchfgeg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad u \in {{\mname{SNoL}}~{({{x}*{y}})}} .  {\unicdcdadad p : {o} .  {{({\unicdcdadad v \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad w \in {{\mname{SNoL}}~{y}} .  {{{{u}+{{v}*{w}}}\unicdcdgded{{{v}*{y}}+{{x}*{w}}}}\unicdbdjdcd{p}}}})}\unicdbdjdcd{{({\unicdcdadad v \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad w \in {{\mname{SNoR}}~{y}} .  {{{{u}+{{v}*{w}}}\unicdcdgded{{{v}*{y}}+{{x}*{w}}}}\unicdbdjdcd{p}}}})}\unicdbdjdcd{p}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 395b74682e5fe85f5a9dd02048bafdd0dde83a774140d2e273cde97e801116d8 \\
Pure Prop Address: & TMLLQmqPzFYwqdwY8PrqVGLuvrDLR95qgbq \\
Theory Prop Id: & ad6f960961622055682865532b9e4b9576d116539cdde8aee2b4846f9ef3628d \\
Theory Prop Address: & TMUjL5DBmrfFRCEH1KtPfFFFPNQ8LHK6fES
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfdfoepgcfpfjgogehfgchahpgmgbgehfg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad u \in {{\mname{SNoR}}~{({{x}*{y}})}} .  {{({\unicdcdaddd v \in {{\mname{SNoL}}~{x}} .  {\unicdcdaddd w \in {{\mname{SNoR}}~{y}} .  {{{{v}*{y}}+{{x}*{w}}}\unicdcdgded{{u}+{{v}*{w}}}}}})}\unicdcdcdid{({\unicdcdaddd v \in {{\mname{SNoR}}~{x}} .  {\unicdcdaddd w \in {{\mname{SNoL}}~{y}} .  {{{{v}*{y}}+{{x}*{w}}}\unicdcdgded{{u}+{{v}*{w}}}}}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e1594953c98071d2fbe62599ce28adefdbe6f718c6c29a167036427051a6c7ae \\
Pure Prop Address: & TMb5PPn6ZNjq3UQXjdnTK8NbeGxGWefm1uY \\
Theory Prop Id: & b67a3574b63d9737488eb186c785d77ae947f04968d568abb2eaa58e430ac851 \\
Theory Prop Address: & TMUGZi51yni6VZeAy2PCP7s6JRSFQbEGocS
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfdfoepgcfpfjgogehfgchahpgmgbgehfgpfjgngahchfgeg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad u \in {{\mname{SNoR}}~{({{x}*{y}})}} .  {\unicdcdadad p : {o} .  {{({\unicdcdadad v \in {{\mname{SNoL}}~{x}} .  {\unicdcdadad w \in {{\mname{SNoR}}~{y}} .  {{{{{v}*{y}}+{{x}*{w}}}\unicdcdgded{{u}+{{v}*{w}}}}\unicdbdjdcd{p}}}})}\unicdbdjdcd{{({\unicdcdadad v \in {{\mname{SNoR}}~{x}} .  {\unicdcdadad w \in {{\mname{SNoL}}~{y}} .  {{{{{v}*{y}}+{{x}*{w}}}\unicdcdgded{{u}+{{v}*{w}}}}\unicdbdjdcd{p}}}})}\unicdbdjdcd{p}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e5cae53a35ffe5b399ffd05d7bd80447303ca0d61aa9feca19313ed6bd26663b \\
Pure Prop Address: & TMYf3j6h6sSAvpn3bX7i2vn59bPSs8V1Lyj \\
Theory Prop Id: & d1fff1afdd85cf834265b58f478397a8d6a7bb46decf4e5fa0e45dfaca997f73 \\
Theory Prop Address: & TMGsDwijjDwd8s7TJ4BtBpPj7TywGA7uPvh
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfkhfgchpgcf} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{x}*{0}}={0}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d279f406aa4b69f7009533d48bdd2cde0236df1ead29105c36976e3dc31e0eea \\
Pure Prop Address: & TMbh4sw15c3nkGcdW9gjqchsqeGnE6CLJka \\
Theory Prop Id: & a4f4a68e9bdee9674a6a2323e1cb9216f16fac58db1b650aa99363d451432e40 \\
Theory Prop Address: & TMX3UwQTHomopN15s51CQVuoHBm4nfFok7n
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfpgogfgcf} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{x}*{1}}={x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1d2f1d16bda75bb4c0ed0799701f296b11ddee1eb905b339cd6220b0faf7d94c \\
Pure Prop Address: & TMPJYXDtPMCDcjFFQegpZzooLCy6nB6AnVt \\
Theory Prop Id: & 5477a4ee853ab494ceac03d98e85ac4588ce7bdd59886ad227b471fce11f4d89 \\
Theory Prop Address: & TMKdX9N9egsSZcy8hkKrdhm8AietaRG3z8M
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfdgpgng} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}*{y}}={{y}*{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 21f7d80766165f191325b0834d9de7e7be3a0dd1d07fa89c1f7a6020d3653a22 \\
Pure Prop Address: & TMWdJKtXqZ53gSQC2BVHdRbKeugBVoakiVX \\
Theory Prop Id: & 83d5fab1574f51f9c204a181c7dfc8be42d76f66ec97dbb3470cdfdc46326b24 \\
Theory Prop Address: & TMdg6XgWdnxnzK7BDdwoHBgq7caePWS9hV1
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfngjgogfhdhpfegjgdhehch} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{({\prenc{{x}}})}*{y}}={\prenc{{{x}*{y}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9beaa01938101baaa11d2398155e5373cc0ae1d7998293c91f378b289f8e122f \\
Pure Prop Address: & TMRSczXWoW4GcLzauYywq4qjgTNRs8DVfKY \\
Theory Prop Id: & 3d0cd1fa56b07fbeb5efae9a27e18be23a0f0160374b403e50685ba3228a818a \\
Theory Prop Address: & TMXe2FHAwQLWs8w3e4pZKSq3Ay1HjVYmAo9
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfegjgdhehchcf} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{({{x}+{y}})}*{z}}={{{x}*{z}}+{{y}*{z}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6a23e513510b3a7b5a3392d09ddfa8bf723376a9ef4220bd708cc688d619d3f6 \\
Pure Prop Address: & TMEse8J6Aed49wCopbm1rWGiRVunCrST9NN \\
Theory Prop Id: & 5f038253af67c067b7090c811a60073e88d35c5f778f03a6091776774636ffef \\
Theory Prop Address: & TMYL91kZXWG9iK5FrgphNrfwHqoEcLWbXTo
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfegjgdhehchme} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}*{({{y}+{z}})}}={{{x}*{y}}+{{x}*{z}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4e7eba7fc4815da2d007f42429b70d6437766813fda9ea365ce905b2c09884fa \\
Pure Prop Address: & TMSEGG8Zev8b3WSFiMRZbgEmXP278T6o11U \\
Theory Prop Id: & 2a483403dad7cd65bfa69608ac938c5560f66186e39f7cfeafc037b7cf113feb \\
Theory Prop Address: & TMXPqQpttxR52gPP1rKLuLFPGP1wExdFugR
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfbgdhdhpgdg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}*{({{y}*{z}})}}={{({{x}*{y}})}*{z}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 71a504fd63097789a8be85f133188fe0e6cdb7c87cd67c75d8f488d15233773b \\
Pure Prop Address: & TMbdjdgGdSBzHFduhV2xBDmPQxmKwQt1tuM \\
Theory Prop Id: & d322c17c41c9815d6f808e736a3c46d7cf138d7fb9de4bfebcd6ce05e597e568 \\
Theory Prop Address: & TMUDFiVX4hFhFSKJNGre9DQ2rsnsdEBsmWG
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfogbgehpfngfhmgpfdfoepg} ${\unicdcdadad n m \in {\mname{omega}} .  {{{{\mname{mul\_nat}}~{n}}~{m}}={{n}*{m}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & da39d396fdbb6dc3abd4d8b61394e2ae8cd83bfda154cafeaa7b0d1ee0288b3b \\
Pure Prop Address: & TMQtAZAzTub1xFgkzJYBwEnD6qz9UzUMmJp \\
Theory Prop Id: & eb633595c5ae47b0fcc4717134102599b1f7892572e59291ab05562597e9fb82 \\
Theory Prop Address: & TMaXHxX9CkqxrxU1kbFsYKxJbdBZWsWsgbd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfjeogpfpgngfghgbg} ${\unicdcdadad n m \in {\mname{omega}} .  {{{n}*{m}} \in {\mname{omega}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7b6023e5c720f7f5082b318b3189f8c277b1498a0c34be6e13516ce9bce0a8f1 \\
Pure Prop Address: & TMbHqwvzDZAcjviA88kYptiX95LF6c3pw8m \\
Theory Prop Id: & 8435a4b44154c9d6a5498ab9f17973fc5f75b3b8098f3884189ec529e738ffc5 \\
Theory Prop Address: & TMLcnoowy4SKdgLinmXu8vq7b1VLk5mBvRB
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:egjgbgegjgdgpfpgahfgog}
We define $\mname{diadic\_open}$ to be ${\uniddcgcg X .  {{{X} \subseteq {{\mname{SNoS\_}}~{\mname{omega}}}}\unicdcdcdhd{\unicdcdadad x \in {X} .  {\unicdcdaddd n \in {\mname{omega}} .  {\unicdcdadad y \in {{\mname{SNoS\_}}~{\mname{omega}}} .  {{{{x}+{\prenc{{{\mname{eps\_}}~{n}}}}}<{y}}\unicdbdjdcd{{{y}<{{x}+{{\mname{eps\_}}~{n}}}}\unicdbdjdcd{{y} \in {X}}}}}}}}}$ of type ${{\iota}\unicdbdjdcd{o}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 16a206dc52b4bb55b4a6c92782a08f06f95014669558c00b452a0ba01b887b1c \\
Pure Object Address: & TMXnCV5QN4u4VL3jUYWi1Q5TKvaYVFiyMAy \\
Theory Object Id: & f8473685b008a719f4b168e65e6b1cd5168f93a2f789ae2758a850f2b196b7db \\
Theory Object Address: & TMFiXJkdoLrmEDAbDLC8bw1WgubyoT6m1rW
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:egjgbgegjgdgpfpgahfgogpfje} ${\unicdcdadad X \subseteq {{\mname{SNoS\_}}~{\mname{omega}}} .  {{({\unicdcdadad x \in {X} .  {\unicdcdaddd n \in {\mname{omega}} .  {\unicdcdadad y \in {{\mname{SNoS\_}}~{\mname{omega}}} .  {{{{x}+{\prenc{{{\mname{eps\_}}~{n}}}}}<{y}}\unicdbdjdcd{{{y}<{{x}+{{\mname{eps\_}}~{n}}}}\unicdbdjdcd{{y} \in {X}}}}}}})}\unicdbdjdcd{{\mname{diadic\_open}}~{X}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 690598e6fd4ea25bb12e3b584dc4b13fe6c60343f666f02f1641e57297d2d891 \\
Pure Prop Address: & TMc3CK4vFU9yhYePJb36m6beBT5HxUzg5Dz \\
Theory Prop Id: & c974cede2afec12b64c094aa129dc1b1d1a692b8afe4e7954cb66d9d2f5c2b77 \\
Theory Prop Address: & TMHeah5uo3Soo2iroPYRB2ntKum4R7fb8uT
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgmepfpgngfghgbg}
We define $\mname{SNoL\_omega}$ to be ${\uniddcgcg x .  {\{y \in {{\mname{SNoS\_}}~{\mname{omega}}}|{{y}<{x}}\}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 5f401f6d84aeff20334d10330efdc9bb3770d636d6c18d803bce11a26288a60d \\
Pure Object Address: & TMS5SZr3PAStrSpayzuJN2wkhBQBnxtsCKv \\
Theory Object Id: & 2e3a675cbb0cd13246446f4a4b5819f9d0cd3a852470bdf73b86e4c9fa5f8aac \\
Theory Object Address: & TMTDoEUQNzUxLj7ehiQCYBDF5vvzDtYJBZ7
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:dfoepgcfpfpgngfghgbg}
We define $\mname{SNoR\_omega}$ to be ${\uniddcgcg x .  {\{y \in {{\mname{SNoS\_}}~{\mname{omega}}}|{{x}<{y}}\}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & d24599d38d9bd01c24454f5419fe61db8cc7570c68a81830565fbfa41321648f \\
Pure Object Address: & TMcZ1fLri43RwrwbvUWagasgXfWF6BJHVSg \\
Theory Object Id: & 255a0c77460ecd33846983d06bff6d5687f5c12703ee773bd7ac29ee80876aee \\
Theory Object Address: & TMZoQpNUd5MbGrEDKDP95Eu98wX9TZSnriY
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:egjgbgegjgdgpfpgahfgogpfdfoepgmepfpgngfghgbgpfje} ${\unicdcdadad z .  {{{\mname{SNo}}~{z}}\unicdbdjdcd{{({\unicdcdadad x \in {{\mname{SNoL\_omega}}~{z}} .  {\unicdcdaddd n \in {\mname{omega}} .  {{{x}+{{\mname{eps\_}}~{n}}}<{z}}}})}\unicdbdjdcd{{\mname{diadic\_open}}~{({{\mname{SNoL\_omega}}~{z}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f034141f72d6c4eb35d80a2119f7efda32b3a2da588de397b25001b9c4cc72d9 \\
Pure Prop Address: & TMZo5wo73TruUc3HVgkyKRi1Lm1wR18VgRA \\
Theory Prop Id: & 7fb6824bab5f2011e10de03e8eea40ef57b1d040bc5a7600c6e49a17598be7f1 \\
Theory Prop Address: & TMKVw1RPWUuG98CDGajHDtbjjBEGV18MFAE
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:egjgbgegjgdgpfpgahfgogpfdfoepgcfpfpgngfghgbgpfje} ${\unicdcdadad z .  {{{\mname{SNo}}~{z}}\unicdbdjdcd{{({\unicdcdadad x \in {{\mname{SNoR\_omega}}~{z}} .  {\unicdcdaddd n \in {\mname{omega}} .  {{z}<{{x}+{\prenc{{{\mname{eps\_}}~{n}}}}}}}})}\unicdbdjdcd{{\mname{diadic\_open}}~{({{\mname{SNoR\_omega}}~{z}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 76200c785beb8c7d3f1ca8ae3c4cf9b46875e0cfc933a9cb2834cf5dfa366c76 \\
Pure Prop Address: & TMVnsa7wcgr6WYawj3FNHQPu1WxbQwjWAQf \\
Theory Prop Id: & 58c46af97ae7d614a76a0d68417ebdde3cae5408341ae09dd687c865366186f0 \\
Theory Prop Address: & TMbr7DWfJLuoLhotsiMYBsLoSh3C8or2K85
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:chfgbgmg}
We define $\mname{real}$ to be ${\{x \in {{\mname{SNoS\_}}~{({{\mname{ordsucc}}~{\mname{omega}}})}}|{{{{{{\mname{SNoL\_omega}}~{x}}\unicdcdgdad{0}}\unicdcdcdhd{{{\mname{SNoR\_omega}}~{x}}\unicdcdgdad{0}}}\unicdcdcdhd{{\mname{diadic\_open}}~{({{\mname{SNoL\_omega}}~{x}})}}}\unicdcdcdhd{{\mname{diadic\_open}}~{({{\mname{SNoR\_omega}}~{x}})}}}\}}$ of type ${\iota}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 1bd1ac4ee8852db6f99d372f1c27ef21bca430b1c9c4c4d184e017f6a4d8d1b4 \\
Pure Object Address: & TMKMHojzxJK5coFqxQsQtHCgmSHKPCW9Vm9 \\
Theory Object Id: & 87ab7c83c1184f9f51c4168a5ec6a5ea9da4a8e2acb1fc82e0f19a7afc1617c4 \\
Theory Object Address: & TMYohVoF9p6nAvkXe1FijCMbejrCeRTmmEN
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:chfgbgmgpfje} ${\unicdcdadad x \in {{\mname{SNoS\_}}~{({{\mname{ordsucc}}~{\mname{omega}}})}} .  {{{{\mname{SNoL\_omega}}~{x}}\unicdcdgdad{0}}\unicdbdjdcd{{{{\mname{SNoR\_omega}}~{x}}\unicdcdgdad{0}}\unicdbdjdcd{{{\mname{diadic\_open}}~{({{\mname{SNoL\_omega}}~{x}})}}\unicdbdjdcd{{{\mname{diadic\_open}}~{({{\mname{SNoR\_omega}}~{x}})}}\unicdbdjdcd{{x} \in {\mname{real}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 09fffc7b9e5d0b1d7f15cd56209324363defbb3ebe9c272e83a66e2590b7bd24 \\
Pure Prop Address: & TMUWdsBKSGjUkuMEbeQNbUqTtaf4Jfjg27G \\
Theory Prop Id: & 594245216a8c68c4e6457b47578cb5437b65d219679a5836cb77b5ff224d6c9a \\
Theory Prop Address: & TMGSVE4T9JHVEamf5z5kbyL8PNooxb94NMt
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:chfgbgmgpffe} ${\unicdcdadad x \in {\mname{real}} .  {\unicdcdadad p : {o} .  {{({{{x} \in {{\mname{SNoS\_}}~{({{\mname{ordsucc}}~{\mname{omega}}})}}}\unicdbdjdcd{{{{\mname{SNoL\_omega}}~{x}}\unicdcdgdad{0}}\unicdbdjdcd{{{{\mname{SNoR\_omega}}~{x}}\unicdcdgdad{0}}\unicdbdjdcd{{{\mname{diadic\_open}}~{({{\mname{SNoL\_omega}}~{x}})}}\unicdbdjdcd{{{\mname{diadic\_open}}~{({{\mname{SNoR\_omega}}~{x}})}}\unicdbdjdcd{p}}}}}})}\unicdbdjdcd{p}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 44069e352f5036a86127073abddc29a0ac5cf4eb6f4d81d6ec3a0209c2755103 \\
Pure Prop Address: & TMHRqA5xCUMEW6aiM8SkmMPejfbXcp8AuW2 \\
Theory Prop Id: & e2acb3ab0b286d7acb20f49882f08eb34a7bbd287abb4ee34541635153010085 \\
Theory Prop Address: & TMbuQMxV1gs1YZpGC9dAoKbCjDcsEGPDtg5
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dffhcgbhpfchfgbgmgpfdfoepgdfpfpgchegdhfhdgdgpfpgngfghgbg} ${{\mname{real}} \subseteq {{\mname{SNoS\_}}~{({{\mname{ordsucc}}~{\mname{omega}}})}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 78ab2f997523a88bf0249dfd2d26b12d3641ee1e1b29b0fe0b6ecbe2bcb9ba5e \\
Pure Prop Address: & TMTqU7cPcHtqdsT73PkgeBYFon4NVibV1rP \\
Theory Prop Id: & 2858a333c0f86c675cb7a2fd1e8d4029fc7b67d5a2e36fcb017031c2a22d4db9 \\
Theory Prop Address: & TMKo5QEXCMn8QagibutbBTzBZSUDAxobgfW
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dffhcgbhpfdfoepgdfpfpgngfghgbgpfchfgbgmg} ${\unicdcdadad x \in {{\mname{SNoS\_}}~{\mname{omega}}} .  {{x} \in {\mname{real}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & db92898b58c469e9ad58b8b3729b507e8cadb7c632fb9757b051ec3b47524b67 \\
Pure Prop Address: & TMJL6KqW8mXtGGFbki5ao4x1Pu9AbFw59T2 \\
Theory Prop Id: & 7315809dd9bd2ef3a3617333ce251afbd32d098f391ab09694db0d0a3ed52701 \\
Theory Prop Address: & TMYg3xroKjVdTtceUrNgH6Jq8ZdKuyS3gD1
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgdefhehafpfdfoepgmepfdfoepgcfpfpgngfghgbg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNoCutP}}~{({{\mname{SNoL\_omega}}~{x}})}}~{({{\mname{SNoR\_omega}}~{x}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 15b221ba0c4ae0e05ed8cc21a7cdf61ac684525faddd1c477ed29fb7cbd9ee7b \\
Pure Prop Address: & TMQ4q89ASduA9c9ftASiNBHz8pQAiMs6fvH \\
Theory Prop Id: & 354d7d3952bf1c176fc5b59a37d8d3fde5ecdce30261e4d7e68896d0d7a6e87a \\
Theory Prop Address: & TMLgzevvcXAVWAzGL5wtnbtt749rGGQR3aV
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgdfpfpgchegdhfhdgdgpfpgngfghgbgpfdfoepgmepfdfoepgcfpfpgngfghgbg} ${\unicdcdadad x \in {{\mname{SNoS\_}}~{({{\mname{ordsucc}}~{\mname{omega}}})}} .  {{x}={{{\mname{SNoCut}}~{({{\mname{SNoL\_omega}}~{x}})}}~{({{\mname{SNoR\_omega}}~{x}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0e7b9203cc682cd4470f749df0d7773bb56f7d49172a681a31039c99f4c6190e \\
Pure Prop Address: & TMHfAHcvWTSrs5RmhM3t5QLf3y1Ps4dbWMu \\
Theory Prop Id: & 1df44e924ecdb42a015d2758e58cb9ae0cb4cf2f67ebce9729fc086bad0fbdd4 \\
Theory Prop Address: & TMHNAhMLb3WzEVoWuDPHTbhjyEcTzN9mZJx
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:chfgbgmgpfdfoepgmepfdfoepgcfpfpgngfghgbg} ${\unicdcdadad x \in {\mname{real}} .  {{x}={{{\mname{SNoCut}}~{({{\mname{SNoL\_omega}}~{x}})}}~{({{\mname{SNoR\_omega}}~{x}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e3f60c2768e445b24135a40107035ad21c03de45130bcda31e71da16caa4c99c \\
Pure Prop Address: & TMKK4pjw5QfzGoVXPooBhrwXjMz5y7zwC83 \\
Theory Prop Id: & 11dba32c06250858d053b157288593cb4e9cc3b5e1cac811216e1643ff3caecb \\
Theory Prop Address: & TMad5zGLRZhwq8fTQPe55TZwskUMePg3kMb
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:chfgbgmgpffgihpfegjgbgegpfmeeh} ${\unicdcdadad x \in {\mname{real}} .  {\unicdcdaddd w \in {{\mname{SNoS\_}}~{\mname{omega}}} .  {{w}<{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8a1a82ac356e457f7d6fea412cc602e7bf973ae5a3d1fe392acbd6183e111060 \\
Pure Prop Address: & TMRmEwRJSZMnW9FxQdrwZVkpY4X3hZtgEsF \\
Theory Prop Id: & 95fe1677ab448626991245b1f6c72063712786eb5fc15e18c8ad861be5a10d24 \\
Theory Prop Address: & TMWnkcpDQgaZXznHwxbP51Zpm4HsdCijkgK
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:chfgbgmgpffgihpfegjgbgegpfheeh} ${\unicdcdadad x \in {\mname{real}} .  {\unicdcdaddd z \in {{\mname{SNoS\_}}~{\mname{omega}}} .  {{x}<{z}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & edbfff85c909fe0b9cc4f471ef0cb3f4b241c89ff5fdc0aac267048126f79d55 \\
Pure Prop Address: & TMWbbSgeg81TLqCRrR2wnZxFJM2VyEnhoKs \\
Theory Prop Id: & 8b82781e914ba17ea6121ff32dd4355d61bc2d5e58cf93f4dfc19f6be51aa543 \\
Theory Prop Address: & TMVen1Z8YEbACtaXxe8EHzxX42Fe1HVxEn9
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfahpgdhpfahpgdh} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{0}<{x}}\unicdbdjdcd{{{0}<{y}}\unicdbdjdcd{{0}<{{x}*{y}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fda2502eea941a875df72ec6015f1570649b018b98f8184cd3c875919ea4099f \\
Pure Prop Address: & TMMBQ62TWFg4vbNhYgkxMEhq8L4wrwpNeH4 \\
Theory Prop Id: & d984dcdb63a86f7c983729aa4499bd351e39b1649325ebc42012cca4a4f9dda6 \\
Theory Prop Address: & TMRy9goqWBW3Lrva8p4Hh4yhTMAF2u9LrbY
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfahpgdhpfogfghg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{0}<{x}}\unicdbdjdcd{{{y}<{0}}\unicdbdjdcd{{{x}*{y}}<{0}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4dec1e722a076d06ad4666c4d9e70af174b210cfaebca7581e05f1d86d354df3 \\
Pure Prop Address: & TMPctCrMmwtqdsDYtwTpmW8cn8ReSGEXW1d \\
Theory Prop Id: & 697e77b4240d0fbd5154af82b65814d654a79f225b48542bc4a3d1b906b094d7 \\
Theory Prop Address: & TMbmWHtMpjybVdPJPF2YcxQPAiySRhKRtU7
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfogfghgpfahpgdh} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}<{0}}\unicdbdjdcd{{{0}<{y}}\unicdbdjdcd{{{x}*{y}}<{0}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6f7ff58caeaa194bad28d270130f11e1db31bb6ab731d5ea0051c78e8fe144e4 \\
Pure Prop Address: & TMHNbtt4BUkPSLXdqKhLABgy4oEcpaoGeQF \\
Theory Prop Id: & 6e7adce5ea23f07343a9f2808aa1618cc2fef1e19abfb8217fb9154a7c79b8f6 \\
Theory Prop Address: & TMX5KmwY5bfi7Ub8rgxQwDCpNPeAkJVtZsN
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfogfghgpfogfghg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}<{0}}\unicdbdjdcd{{{y}<{0}}\unicdbdjdcd{{0}<{{x}*{y}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 86d604810bc69f81920bb16b33d0acc3bedea0cfb10d12b4458a7d1fafe8a9d9 \\
Pure Prop Address: & TMaUyk4nPJDDrgihhHeRFKvQ2fABVtnDW2a \\
Theory Prop Id: & 03517a8b5c20769332200cc5606cad376758ee52bb6086d3523cb789f19cee77 \\
Theory Prop Address: & TMZsT4fPodxbAqp3EhReFUneq4rHkVaHEZS
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngfhmgpfdfoepgpfogpgogkhfgchpg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}\unicdcdgdad{0}}\unicdbdjdcd{{{y}\unicdcdgdad{0}}\unicdbdjdcd{{{x}*{y}}\unicdcdgdad{0}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 62b0712aafda95c0ed43facad05fea37b6fb484147b8453d916f90fe017d591f \\
Pure Prop Address: & TMGU71yz2XNvrMWpSrZ2VaZAXfAZoXYkpsg \\
Theory Prop Id: & edd14a5db38e09a96214c426f3ca626167e570af71109dca57dc26ce0cdfae63 \\
Theory Prop Address: & TMH1iV5T59LgjrgGemBxMfYGnwBeTHPSpxr
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfchfgdhehchpfdfoepg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{\unicdcdadad alpha \in {{\mname{SNoLev}}~{x}} .  {{{({\prenc{{x}}})}\unicdcdcdjd{{\mname{SNoElts\_}}~{\alpha}}}={\prenc{{({{x}\unicdcdcdjd{{\mname{SNoElts\_}}~{\alpha}}})}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6e2d93e0dc17abea886b929fe5995e073d3fdcf446b9f988050524e8d71bd48f \\
Pure Prop Address: & TMG933hdiaf584T7p9ZFRvPn9oiCEWieGQW \\
Theory Prop Id: & 33a9b7427401c079a1d2742931e0769dcffafc114bd1bf1548f1b32dbd6c8dc6 \\
Theory Prop Address: & TMYmCxYkd92jjewaiC7sVwLTus2QgwYLten
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpffgihbgdgehmgjhbdpgggcd} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{\unicdcdadad alpha \in {{\mname{SNoLev}}~{x}} .  {{{\mname{exactly1of2}}~{({{\alpha} \in {x}})}}~{({{\alpha} \in {\prenc{{x}}}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8867febe718634f83190f3f8c3dab70b7c761d564ddd1acb9cadda95bf892f1b \\
Pure Prop Address: & TMbbMHfSdx97M7MMua7mnYgvKSs2grfdonk \\
Theory Prop Id: & c2e4600c71315f1fd85398873234adada26e385afa312ce802aceb4cd7e1c2b3 \\
Theory Prop Address: & TMZRH9eUQQW9BQLZhXvbWDnscfgzh7sfosN
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfjeog} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{\unicdcdadad alpha \in {{\mname{SNoLev}}~{x}} .  {{{\alpha} \in {x}}\unicdbdjdcd{{\alpha}\unicdcdadjd{\prenc{{x}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 634ffccfb2da872d1912604afc7221a4c2f97936c287ab1c4253071e1dfcfc61 \\
Pure Prop Address: & TMRJdHvUjKuqbmun53qwo4QgnKCJpLgQbyt \\
Theory Prop Id: & 6ad85181a4e76fe3485b7a68f6c450251768534fcb5c6074cb42e2acf9661987 \\
Theory Prop Address: & TMHfvPTQWHBMA7BghiU2NwAbmLn8jDpkF9g
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfogjeog} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{\unicdcdadad alpha \in {{\mname{SNoLev}}~{x}} .  {{{\alpha}\unicdcdadjd{x}}\unicdbdjdcd{{\alpha} \in {\prenc{{x}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 42b82e1e669aebbf602bf5113f1d414db866cfe089bda3a8f91d980976cd7959 \\
Pure Prop Address: & TMcrVxUeXZKH9hKHxpogFheX7RW2QbWzvcH \\
Theory Prop Id: & c24e982ace7dff228ddfbb28d50b4cddc7e5842b4c70d4190e10a022c64bebc0 \\
Theory Prop Address: & TMbXvwL69SpXNMdrP7cnTzE8ykxUg2LWeVZ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:chfgbgmgpfngjgogfhdhpfdfoepg} ${\unicdcdadad x \in {\mname{real}} .  {{\prenc{{x}}} \in {\mname{real}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5ba504335f94f103b4e4f4b4128f81159e7d57c6f6e51ddf1aac116c6bc3d844 \\
Pure Prop Address: & TMRu8g6RVBXBFKY32BG4WQRsE2JJUCAdD8A \\
Theory Prop Id: & 54413c793a2a6cf0fcb3da3a35724c098c7e8b441474a0aac138fbd28ee3cb5c \\
Theory Prop Address: & TMJ5SdaCycBTmAK8QT7hmjqUxcKbkFRkZSH
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:egjgghpfdfoepg}
We define $\mname{div\_SNo}$ to be ${\uniddcgcg x y .  {\mif{{{y}={0}}}{{0}}{{{\mname{Eps\_i}}~{({\uniddcgcg z .  {{{\mname{SNo}}~{z}}\unicdcdcdhd{{{z}*{y}}={x}}}})}}}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 16510b6e91dc8f8934f05b3810d2b54c286c5652cf26501797ea52c33990fa93 \\
Pure Object Address: & TMJtKhWrCvxZo4EySoUGxRLHGe8kxsrf5wV \\
Theory Object Id: & 41b026a24f3f123a0f6f2e71ea4d39925a2b48bf6dedd1c902de3b86264323f1 \\
Theory Object Address: & TMXu1QtbM1e9R4bcCcqrQqVE2nbX4D1QdbG
\end{tabular}
\end{center}

\end{definition}
{\bf{Notation.}} We use $\infkdpckd$ as an infix operator  corresponding to applying term ${\mname{div\_SNo}}$.
\begin{definition}\label{def:fgihahpfdfoepgpfogbgeh}
We define $\mname{exp\_SNo\_nat}$ to be ${\uniddcgcg n m : {\iota} .  {{{{\mname{nat\_primrec}}~{1}}~{({\uniddcgcg \_ r .  {{n}*{r}}})}}~{m}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & cc51438984361070fa0036749984849f690f86f00488651aabd635e92983c745 \\
Pure Object Address: & TMHssmmer5vtejxULVomtdegStRPMUKFbiM \\
Theory Object Id: & ea81040eaa779da7e35317f42e478d807fb38e0dd71814599673b64763a57fba \\
Theory Object Address: & TMPrqwjmsGkTo9k9ojpFo3twgRABMuqrPhD
\end{tabular}
\end{center}

\end{definition}
{\bf{Notation.}} We use $\infof$ as a right associative infix operator  corresponding to applying term ${\mname{exp\_SNo\_nat}}$.
\section{Complex Surreals} % SurrealArithmetic
\begin{definition}\label{def:dedfoepg}
We define $\mname{CSNo}$ to be ${\uniddcgcg z .  {\unicdcdaddd x .  {{{\mname{SNo}}~{x}}\unicdcdcdhd{\unicdcdaddd y .  {{{\mname{SNo}}~{y}}\unicdcdcdhd{{z}={{{\mname{SNo\_pair}}~{x}}~{y}}}}}}}}$ of type ${{\iota}\unicdbdjdcd{o}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & c35281fa7c11775a593d536c7fec2695f764921632445fa772f3a2a45bdf4257 \\
Pure Object Address: & TMUD5RtiGmKRpxwvJihKA6DLDFMXAMhjUgo \\
Theory Object Id: & 9f6440ba5a12befd32f7a9856b6769fa208407cf0c9b4747cb6036bb0d6250a3 \\
Theory Object Address: & TMLizygxxxRc6MNrrbecR4regbF9SnDnoL1
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:dedfoepgpfje} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{\mname{CSNo}}~{({{{\mname{SNo\_pair}}~{x}}~{y}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a092484c55b8291c75d0d36b92ada8b7272277bdbed2f8e85dd4647982d82a11 \\
Pure Prop Address: & TMLuqWx8Kn7Zr4K7kCY3vBtreAy6N2Gq5Ho \\
Theory Prop Id: & b22a0800c64b20ca0a19ed9cf3cf1103cba24b4506c039ffdc78edc941d716bc \\
Theory Prop Address: & TMSb6wFZKXobEGqJEzsAV83ZDCU7SEHmi81
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dedfoepgpffe} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{({\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{z}={{{\mname{SNo\_pair}}~{x}}~{y}}}\unicdbdjdcd{{p}~{({{{\mname{SNo\_pair}}~{x}}~{y}})}}}}}})}\unicdbdjdcd{{p}~{z}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d92da0243181c2c853df14146584be693f357edce1fe0d6dc91b3a43175cd9a5 \\
Pure Prop Address: & TMF2YxYA3n45UKAAPKZdwQUZ5pkopFy84Zo \\
Theory Prop Id: & fdbc8d36959b5172de4f1d46ecd65e45290e18a238b0d3274dd0238e86f92e5a \\
Theory Prop Address: & TMREupmRGKvMjoTNRuLkffjgfQYrtqyaxmd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfdedfoepg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{\mname{CSNo}}~{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 56121e05e33db2f68607348644f875d4c59e36314e4c7aade2ed7acf3bc097ea \\
Pure Prop Address: & TMKSgGTcxX9dqXoEQ6Z8PkWivenDVtKxQTL \\
Theory Prop Id: & d50db589d56adb9c8fd5479df494050c31aaa3429ec352b511a7b0a13aeea945 \\
Theory Prop Address: & TMR4DcyLohCpAbSXv9WH4Xcg5MLahA6HKCW
\end{tabular}
\end{center}

\end{theorem}
\section{Complex}\label{depgngahmgfgih}\begin{definition}\label{def:depgngahmgfgihpfjg}
We define $\mname{Complex\_i}$ to be ${{{\mname{SNo\_pair}}~{0}}~{1}}$ of type ${\iota}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & d0c55cfc8a943f26e3abfa84ecab85911a27c5b5714cd32dcda81d104eb92c6e \\
Pure Object Address: & TMRLLQsBvKWihFburT51dAV1GsLtPYwc9Zk \\
Theory Object Id: & df0e4e103bfa2441c4757b4d5b0e56da78eeec142d6eb59cb57bf6f96d041b2e \\
Theory Object Address: & TMWUiSAowU8u6wDqUE8abxMdTuajU7UtstH
\end{tabular}
\end{center}

\end{definition}
Let $i$ be ${\mname{Complex\_i}}$.
\begin{theorem}\label{thm:dfoepgpfdepgngahmgfgihpfjg} ${{\mname{CSNo}}~{i}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0b8add253d6a140458980890174b902430843878d8fbe0e4e45711dca51045e1 \\
Pure Prop Address: & TMTBCkaSu5M84GaWqNVMA2i4e55BjcK87XM \\
Theory Prop Id: & 139a531c852859624509b40bb3d55cf6dc1c47b1c91e4e56557dd5eca9352643 \\
Theory Prop Address: & TMMieXaSc5pjuKyowxbkxfTqbz2F6SwKDZS
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dedfoepgpfcffg}
We define $\mname{CSNo\_Re}$ to be ${\uniddcgcg z .  {{\mname{Eps\_i}}~{({\uniddcgcg x .  {{{\mname{SNo}}~{x}}\unicdcdcdhd{\unicdcdaddd y .  {{{\mname{SNo}}~{y}}\unicdcdcdhd{{z}={{{\mname{SNo\_pair}}~{x}}~{y}}}}}}})}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 9481cf9deb6efcbb12eccc74f82acf453997c8e75adb5cd83311956bcc85d828 \\
Pure Object Address: & TMbtVDb4DspXgdVe6YzPV3kDEEEwHY6ZCot \\
Theory Object Id: & 78e2fa14ee2f6baf7a5e87660241882e064e475595b0b4d3abb32b22e16d4844 \\
Theory Object Address: & TMcoCaQ7XUy1bDRwyD8B1Hc9XvedDZEaj9Z
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:dedfoepgpfjeng}
We define $\mname{CSNo\_Im}$ to be ${\uniddcgcg z .  {{\mname{Eps\_i}}~{({\uniddcgcg y .  {{{\mname{SNo}}~{y}}\unicdcdcdhd{{z}={{{\mname{SNo\_pair}}~{({{\mname{CSNo\_Re}}~{z}})}}~{y}}}}})}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 5dad3f55c3f3177e2d18188b94536551b7bfd38a80850f4314ba8abb3fd78138 \\
Pure Object Address: & TMZBTXQFnkALw8eLrVDkBW9kDUBUNtJ9KV1 \\
Theory Object Id: & 995ec6771de10483744c42c2860b06ba0be54c9d0335cccb1a962e90378575b5 \\
Theory Object Address: & TMFEq3qfdWzjBymvGvoarLmQpoH3fgwe9yD
\end{tabular}
\end{center}

\end{definition}
Let $Re : {{\iota}\unicdbdjdcd{\iota}}$ be ${\mname{CSNo\_Re}}$.
Let $Im : {{\iota}\unicdbdjdcd{\iota}}$ be ${\mname{CSNo\_Im}}$.
Let $pa : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ be ${\mname{SNo\_pair}}$.
\begin{theorem}\label{thm:dedfoepgpfcffgbd} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{({{Re}~{z}})}}\unicdcdcdhd{\unicdcdaddd y .  {{{\mname{SNo}}~{y}}\unicdcdcdhd{{z}={{{pa}~{({{Re}~{z}})}}~{y}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d1afc4d126dd465be260002c44748dfd6b52c42ef68b99e2aca9d39589415eff \\
Pure Prop Address: & TMVD9LSTuGahX5mYEQHbkD3Aj4r6YjvDqav \\
Theory Prop Id: & dda4fdb9868a65730eda625dcc1b25d74a368899640ab43b19a56dee292fd951 \\
Theory Prop Address: & TMdw63sz9FuEG3u34nTUC9xRWJSnjKoK5iM
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dedfoepgpfcffgcd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{Re}~{({{{pa}~{x}}~{y}})}}={x}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b2daba3d81b7c6ff9fbeddbdc673af753aea0654c6f0beb2e4150cec6aa3ff36 \\
Pure Prop Address: & TMbd79jCDur1j41vTRXKVgXL6tDG3dczCch \\
Theory Prop Id: & c4a4431b7dc74e1c59f2d3f78882e9511898bb365f4441eb504d35a102c73f5d \\
Theory Prop Address: & TMTPxuYePSpdrjC2hZR9YA4mNhhCM2obwvy
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dedfoepgpfjengbd} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{{\mname{SNo}}~{({{Im}~{z}})}}\unicdcdcdhd{{z}={{{pa}~{({{Re}~{z}})}}~{({{Im}~{z}})}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 74e0f37f75ccb3555e54f4af34c332761bbf40f68215121dd3426a5fc1d57563 \\
Pure Prop Address: & TMbMFKS5usWyvTh28krP3yJ58GAo85ggE5u \\
Theory Prop Id: & f920bd6c9ae2ef46e0da84205614a41dca1aa86484bc96ed97046e5cb2dcc706 \\
Theory Prop Address: & TMSHmgJJG9JSYax4SjR2fRxfC1xbTdvaFHp
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dedfoepgpfjengcd} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{Im}~{({{{pa}~{x}}~{y}})}}={y}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2bc6171b4c1d4c23d2b8257ee568deebce1bdeeb5abadafad24f46402844bec8 \\
Pure Prop Address: & TMPfMNHVuYrdTndLNgbw7ssKmmTUSmVQhuq \\
Theory Prop Id: & 8be49e5c0e40a6664121bff48d6f103b94ded1a85e8276f4270b2fecc4365f2d \\
Theory Prop Address: & TMUySj1Kdt4rWdt8CvGuPiFauFwdwqGuBrt
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dedfoepgpfcffgcf} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{\mname{SNo}}~{({{Re}~{z}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2d8bafb4716aba45ba78940b437d476181b1832c88e0fdf73d1f44139d745aa6 \\
Pure Prop Address: & TMHWBdpBEx7e2XaqsRv4PexZFWK6mxntB5L \\
Theory Prop Id: & 9eab1faa688e3e8a126798d1a4feec2d230170ca2c79e1f6a7cac4413ce087e6 \\
Theory Prop Address: & TMLjD3HKc5VtY2zV8JTsHHEn8qrJqupKsD9
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dedfoepgpfjengcf} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{\mname{SNo}}~{({{Im}~{z}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ab80bc51455c8401ea6cfc9fb8b50fce5202d296b70813e18ab3b4578724d01b \\
Pure Prop Address: & TMHM4TWxDfhFey1F4K4ZfTkNgJ3aXPsHVnx \\
Theory Prop Id: & f91d5d3724c3d6ffc90c6787c4369da0ef974adfbd2841122b8eea32364c3543 \\
Theory Prop Address: & TMaeNJLGGokLzoVMSdnkmyBJB7tx3aSEAk3
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dedfoepgpfcffgjeng} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{z}={{{pa}~{({{Re}~{z}})}}~{({{Im}~{z}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 445a11e5d038263fe95c6356046a0e1e52288c7a48a5d5c22900215dc15818bf \\
Pure Prop Address: & TMbfkg4MdezPBN2Y4Qjkj5qcWqmWQ3YK3MM \\
Theory Prop Id: & 520c8f693aef23e47d75aa0230b2dc952db5beee2eec2007112e6239da337856 \\
Theory Prop Address: & TMPQ5WLn58Hodm558JuPKepWo3qKYLgWAfo
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dedfoepgpfcffgjengpfdhahmgjgeh} ${\unicdcdadad z w .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{{\mname{CSNo}}~{w}}\unicdbdjdcd{{{{Re}~{z}}={{Re}~{w}}}\unicdbdjdcd{{{{Im}~{z}}={{Im}~{w}}}\unicdbdjdcd{{z}={w}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1b6f8a62f72e786e4c9bf9f10ef1b663021c7dd462f9001118924d0c3eb05d56 \\
Pure Prop Address: & TMF22319Si8r5cP9dtLLXWhca498DPYaPfW \\
Theory Prop Id: & e59d00e026008d1f4bb97a82ee1e249a3a6533f81c95228512a948d73a5948a1 \\
Theory Prop Address: & TMQQTkXaTqRkMfBzratppu8SFd9kJhSw1K9
\end{tabular}
\end{center}

\end{theorem}
{\bf{Notation.}} We use $\prenc{-}$ as a prefix operator  corresponding to applying term ${\mname{minus\_SNo}}$.
{\bf{Notation.}} We use $\inflc$ as a right associative infix operator  corresponding to applying term ${\mname{add\_SNo}}$.
{\bf{Notation.}} We use $\infkc$ as a right associative infix operator  corresponding to applying term ${\mname{mul\_SNo}}$.
\begin{definition}\label{def:ngjgogfhdhpfdedfoepg}
We define $\mname{minus\_CSNo}$ to be ${\uniddcgcg z .  {{{pa}~{({\prenc{{{Re}~{z}}}})}}~{({\prenc{{{Im}~{z}}}})}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 9c138ddc19cc32bbd65aed7e98ca568d6cf11af8ab01e026a5986579061198b7 \\
Pure Object Address: & TMLB9tLVVqyKeYKxjvFwaspknDz9iJsXi8C \\
Theory Object Id: & e44642b3fe572281d00e78cdd9551e412cd2f9c70b5057cc266234cae61b142c \\
Theory Object Address: & TMcRv5uzwYDFcF46XsKgFNN7RdY9tuZbeEK
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:bgegegpfdedfoepg}
We define $\mname{add\_CSNo}$ to be ${\uniddcgcg z w .  {{{pa}~{({{{Re}~{z}}+{{Re}~{w}}})}}~{({{{Im}~{z}}+{{Im}~{w}}})}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 30acc532eaa669658d7b9166abf687ea3e2b7c588c03b36ba41be23d1c82e448 \\
Pure Object Address: & TMEg3b8dv5i7oFXmXe8scTrFUcM8PzJ4zpW \\
Theory Object Id: & b095bc500f522f93df919a70a2f70782f5980aa96ce262ebe8358b56594a3c2f \\
Theory Object Address: & TMaA7uRAA9rzybfgoRTMJhUqq2bqKoX4Wx4
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:ngfhmgpfdedfoepg}
We define $\mname{mul\_CSNo}$ to be ${\uniddcgcg z w .  {{{pa}~{({{{{Re}~{z}}*{{Re}~{w}}}+{\prenc{{({{{Im}~{z}}*{{Im}~{w}}})}}}})}}~{({{{{Re}~{z}}*{{Im}~{w}}}+{{{Im}~{z}}*{{Re}~{w}}}})}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & e40da52d94418bf12fcea785e96229c7cfb23420a48e78383b914917ad3fa626 \\
Pure Object Address: & TMG6o42Tv7okyH5PRhCzCahxTbhTrJK9kr3 \\
Theory Object Id: & 9567b4289023a19d5a345a5792fabad48f116c4488984243ea731c878bfe55aa \\
Theory Object Address: & TMKcJrsKgvdh8ShstdBRhZw6Tr7gLxdNL7q
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:egjgghpfdedfoepg}
We define $\mname{div\_CSNo}$ to be ${\uniddcgcg x y .  {\mif{{{y}={0}}}{{0}}{{{\mname{Eps\_i}}~{({\uniddcgcg z .  {{{\mname{CSNo}}~{z}}\unicdcdcdhd{{{{\mname{mul\_CSNo}}~{z}}~{y}}={x}}}})}}}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 98e51ea2719a0029e0eac81c75004e4edc85a0575ad3f06c9d547c11b354628c \\
Pure Object Address: & TMcjyPavg5d3bDQmhU1GhdJgnZiMVguCn1Q \\
Theory Object Id: & 144bb6125865eeb3fbeec27048fa8deb33d515116b6f667efc6200b817c7e681 \\
Theory Object Address: & TMTPDUmndBUSQQFuYNMpTVmCmdY9FA3BZtX
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:dedfoepgpfngjgogfhdhpfdedfoepg} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{\mname{CSNo}}~{({{\mname{minus\_CSNo}}~{z}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7ee6ad0a8aa89406d6b84b29278fb1765127152d723155b4a1c9d6f6c0fea591 \\
Pure Prop Address: & TMKJwYdVTJY7k1e8hMuzVYgoKvGEEEBByvU \\
Theory Prop Id: & 309c1df653ed60931ce0009531575a033315729a35f70a72363bc4dc5a0b3d22 \\
Theory Prop Address: & TMFr3QPLTA3mRgec65aLawX7v5PhREet73h
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfcffg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{Re}~{x}}={x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 62c082375447ec523dbbb2ba00907a038d5e29d38b336bb7ebcfe5b9350e5b40 \\
Pure Prop Address: & TMUR6cn1xni6GYWDsvMbvTJXUokaMMtL4U4 \\
Theory Prop Id: & b4d75e236f52950d7d4585f33566679757107a6a43067a557b1b954b9c81d095 \\
Theory Prop Address: & TMcerjrejE4PH4juW3mMCPrs95Q3KqvPrwh
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dfoepgpfjeng} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{Im}~{x}}={0}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 79fa36b0f877aa0ebc7f2b10cd692fd78dbc9d6316fc4e53d0c61bab998ecf30 \\
Pure Prop Address: & TMZ4xEKBB6n896d4JKvevEKTFR67E7xpLGS \\
Theory Prop Id: & 045ef70364115765ca4038bb6cadac8b50b511da827132c78a50ec9ae776e935 \\
Theory Prop Address: & TMGyhJYv437WcR94jiMRHMwX2AwFbxWRWtp
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgpfad} ${{{Re}~{0}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b67351949a46cdf795a5f976b27f975352929b68d5d5087280de8114752abcf2 \\
Pure Prop Address: & TMU3hZhd9p6dUuXdcCT5sFV8f9rU1SLuLvg \\
Theory Prop Id: & 4d3237f583747a05847601b9321cb843fa14501467c9f468a5a30405685ab1cf \\
Theory Prop Address: & TMPVj3k4kQSu82UN1JnuwzTMbxZYQkycJGG
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:jengpfad} ${{{Im}~{0}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9c8b687bb8d2707b766077cc814cd6418fa32bda09f73340d3b7538ba73166a2 \\
Pure Prop Address: & TMQ5sK61iFZnh5NvkYvBS6MNzDSzr4Q2mQx \\
Theory Prop Id: & c115486aac15de92b350f5674a5898e90fd8a9590b7695815ace4de38d425d98 \\
Theory Prop Address: & TMR2c3XQQEW8f7iwU4myNJae1rZMRsRSBpo
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgpfbd} ${{{Re}~{1}}={1}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 88a2aa81be9b1d2f724a88816564d021c38a577fdbf421180a4c9a6b8d80afc7 \\
Pure Prop Address: & TMdGDE8JarQ1jbtW4A8oVRGior9g3VHcxwo \\
Theory Prop Id: & a28b943c82a02609fdf4f3d340fd99c6f99f7fd4e1f24cf07c819165b82ba877 \\
Theory Prop Address: & TMR387PaXEC5B2kz1ZyEUXJc9eNwtkfqgH2
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:jengpfbd} ${{{Im}~{1}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3b8fb5faff6bf916e912016e794c0366b38e4279d8fe1cdeeee5aa0afc03878d \\
Pure Prop Address: & TMZBLCABN62himZDekvtrQPidyv7fcLegSk \\
Theory Prop Id: & 1ca93cd098ccdc1ee3d42ee3ff16671c864fc5e68c4548d64594dc3f72490d3f \\
Theory Prop Address: & TMXR9fNVoFqUzmgKeV3ME2ptYRiTSH3E7yW
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgpfjg} ${{{Re}~{i}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8f12f0decf716ca445164b75da58be3f2a7555cb429371632eb027a71508c1d1 \\
Pure Prop Address: & TMK96CbwKZMULThVCgvnQBDjGHq6LXfbzfA \\
Theory Prop Id: & bfd076483d6cacc95ee0f7ff33c3ceda810e4fef403a422c0ecef2104636ffdf \\
Theory Prop Address: & TMWVmowUnC9iG9jmJkW8uaTKvWjDDPGhJaq
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:jengpfjg} ${{{Im}~{i}}={1}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1795ca8f39eb6b9aa0312ca4743f676b7dad188738ed6523cb5b0e265558330b \\
Pure Prop Address: & TMUhvnCxhioSCB7KmHbHrUrhDXFyjVzSfy7 \\
Theory Prop Id: & 6a700bf595245199596fb74a524dc20a87be2fa234d75a9425a486afde75e739 \\
Theory Prop Address: & TMV23ugJNe6xxnjymVV5jwq448H5U6cUMNz
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdfoepgpfbgegegpfdedfoepg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}+{y}}={{{\mname{add\_CSNo}}~{x}}~{y}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7f45ee359a31d432da4768def1fc35bea04f7baa2682362d567be00cc83e019c \\
Pure Prop Address: & TMPEXTCLw1a9KLNu4LjTEYeAiXpBwtm8LYN \\
Theory Prop Id: & a9fb7d4440806134531895d72ae0d8c65453fa88d1b4a12beeccef25e3784bcc \\
Theory Prop Address: & TMYERyzMgccEbP9urACTzpKQZ6NqHoJHCpt
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dedfoepgpfbgegegpfdedfoepg} ${\unicdcdadad z w .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{{\mname{CSNo}}~{w}}\unicdbdjdcd{{\mname{CSNo}}~{({{{\mname{add\_CSNo}}~{z}}~{w}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e57a7792a9adb12eaa7d6f4b276a0d7c46de087f9e650c63fa726890a32a5ea5 \\
Pure Prop Address: & TMTfa7wPMhSgYfANj3ojEMJAqmSXEbR5wUi \\
Theory Prop Id: & 9f039b5cd4b098a77ad130de4441f49110a2a2055cf0ea279af6db8406a33f15 \\
Theory Prop Address: & TMRZouHFioSkxKnrxrt5ocqpagQGyQhPBxr
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdedfoepgpfadme} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{{{\mname{add\_CSNo}}~{0}}~{z}}={z}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e96f34c9cab6bff68069bd60d532d81645e33b9518db1f62bf0603eb2cb2b881 \\
Pure Prop Address: & TMVPvy3qhQSp4Z8EZHLGefCshoKUigGYBXD \\
Theory Prop Id: & 9130336e1f65fce4abcbac21206949f4a80b64433ce8b55a026e39164bbe17ae \\
Theory Prop Address: & TMJn8298DRnHjSHEgaMHnaaNn1aon8VyNao
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdedfoepgpfadcf} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{{{\mname{add\_CSNo}}~{z}}~{0}}={z}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 013f9d9fcb6813d280a54b0c238a0259bbf245f8c19e895056f52e4c3adeb31e \\
Pure Prop Address: & TMMjUSCaFbW2xfSEQbTD6ikQsQfwYk4GFkA \\
Theory Prop Id: & c676dd1073460165d7f272c397af95f64013350b90182b6212b134b88768602a \\
Theory Prop Address: & TMSwnX5gYRWnQPynxcvBXkQVYgYNF2643Z5
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdedfoepgpfngjgogfhdhpfdedfoepgpfmgjgoggh} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{{{\mname{add\_CSNo}}~{({{\mname{minus\_CSNo}}~{z}})}}~{z}}={0}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 87ef9663f32d233b6c1ad44f942d450a76582c2e2ac1a28a691747d6debc0bb4 \\
Pure Prop Address: & TMU75d85SiBY1nyEa7XDpioTuTb27YfaBGF \\
Theory Prop Id: & 08935706a424eeef4a646b98c7dc667515cfe98f6a4e99488cf85e717d7bf96d \\
Theory Prop Address: & TMWn3sP7p5YyoTSSYF8fydsrBPLkb9tjRKH
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:bgegegpfdedfoepgpfngjgogfhdhpfdedfoepgpfchjgoggh} ${\unicdcdadad z .  {{{\mname{CSNo}}~{z}}\unicdbdjdcd{{{{\mname{add\_CSNo}}~{z}}~{({{\mname{minus\_CSNo}}~{z}})}}={0}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 350994f849f5d91eeb37d84462edfab4a847176479ae191377608bb65c929618 \\
Pure Prop Address: & TMRirTWCrP5cVCz5MNiHfJ1PfNDmQMnbMjY \\
Theory Prop Id: & 268ed1aa6845aff2a06bc069ee87b3e46e06fa4ca30e8fe25c18f5b8541b8d69 \\
Theory Prop Address: & TMH6YAu7Qh6P2UjC47FxTcP6vhxmhZgf3V8
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ngjgogfhdhpfdfoepgpfngjgogfhdhpfdedfoepg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{\prenc{{x}}}={{\mname{minus\_CSNo}}~{x}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 95399dcec725f3eaf0d48ead791b18326af59f257c823337fdccdeb08631a78c \\
Pure Prop Address: & TMc7tz6FA1Zoz96tsmZ9t1mi6UymeB4f5gM \\
Theory Prop Id: & 23ad035b79e39c7c63924d15360f357008dcb6ef5d26f79c98ed7ee083b32498 \\
Theory Prop Address: & TMYnJa2FrK92ftPPxTYDAiWjQqyzmtH1euw
\end{tabular}
\end{center}

\end{theorem}
\section{Complex II}\label{depgngahmgfgih}{\bf{Notation.}} We use $\prenc{-}$ as a prefix operator  corresponding to applying term ${\mname{minus\_CSNo}}$.
{\bf{Notation.}} We use $\inflc$ as a right associative infix operator  corresponding to applying term ${\mname{add\_CSNo}}$.
{\bf{Notation.}} We use $\infkc$ as a right associative infix operator  corresponding to applying term ${\mname{mul\_CSNo}}$.
{\bf{Notation.}} We use $\infkdpckd$ as an infix operator  corresponding to applying term ${\mname{div\_CSNo}}$.
\begin{definition}\label{def:jgogeh}
We define $\mname{int}$ to be ${{\mname{omega}}\unicdcdcdbg{\{{\prenc{{n}}}|n \in {\mname{omega}}\}}}$ of type ${\iota}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 4daffb669546d65312481b5f945330815f8f5c460c7278516e497b08a82751e5 \\
Pure Object Address: & TMTGgpvNpXMeoR3NwZFd8y2xpBzTUSKgwbU \\
Theory Object Id: & 50d206e99485ce72d7138ea33476258564bafcdd8979a0e0edf31a56964bb534 \\
Theory Object Address: & TMYfYSNfpYrjF2bVUtReGwX4aiKctLkRVZM
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:chbgehjgpgogbgmg}
We define $\mname{rational}$ to be ${{{{{\mname{ReplSep2}}~{\mname{int}}}~{({\uniddcgcg \_ .  {\mname{omega}}})}}~{({\uniddcgcg num den .  {{den}\unicdcdgdad{0}}})}}~{({\uniddcgcg num den .  {{num}:/:{den}}})}}$ of type ${\iota}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 604584444cba0be7f74a050669bc3716573bcc6798d790d2a2e1d8203a2fd4d1 \\
Pure Object Address: & TMHgCpKhfWk6RuSojtz9kE5WKRRiXLgKQKG \\
Theory Object Id: & cab0d274c7daec544f49542150159453f1d3ebe31677ddce588712380bdb3ffb \\
Theory Object Address: & TMJ8qVVQVeuLH1QKNaSQ3YvEGoSB176XS43
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:dffhng}
We define $\mname{Sum}$ to be ${\uniddcgcg m n f .  {{{{\mname{nat\_primrec}}~{0}}~{({\uniddcgcg k r .  {\mif{{{k} \in {m}}}{{0}}{{{{f}~{k}}+{r}}}}})}}~{({{\mname{ordsucc}}~{n}})}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{\iota}})}\unicdbdjdcd{\iota}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & d8ee26bf5548eea4c1130fe0542525ede828afda0cef2c9286b7a91d8bb5a9bd \\
Pure Object Address: & TMFy5sQDCD3u6eVG9houLmF2KaGQyzqJsCp \\
Theory Object Id: & 81ddb08445e3612684413209359bef93d8b0cf5c8394caeca364d20434961abd \\
Theory Object Address: & TMcbn2wMDtNVYfjf2vBVxsgPGPuUtbFsxBs
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afchpgeg}
We define $\mname{Prod}$ to be ${\uniddcgcg m n f .  {{{{\mname{nat\_primrec}}~{1}}~{({\uniddcgcg k r .  {\mif{{{k} \in {m}}}{{1}}{{{{f}~{k}}*{r}}}}})}}~{({{\mname{ordsucc}}~{n}})}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{\iota}})}\unicdbdjdcd{\iota}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & db3d34956afa98a36fc0bf6a163888112f6d0254d9942f5a36ed2a91a1223d71 \\
Pure Object Address: & TMdN8KB95hDDy7J7CZ25zug8XWiYBFRqyCp \\
Theory Object Id: & da66d8ac9780f328f37f0b84d5376c156ffd8cf8b9eaf38731c6ac83d286cb0e \\
Theory Object Address: & TMSX8CAGQArDkpR8V6Cb73nTcPCTkLUYXtA
\end{tabular}
\end{center}

\end{definition}
\section{Int}\label{jeogeh}{\bf{Notation.}} We use $\inflc$ as a right associative infix operator  corresponding to applying term ${\mname{add\_SNo}}$.
{\bf{Notation.}} We use $\infkc$ as a right associative infix operator  corresponding to applying term ${\mname{mul\_SNo}}$.
{\bf{Notation.}} We use $\prenc{-}$ as a prefix operator  corresponding to applying term ${\mname{minus\_SNo}}$.
{\bf{Notation.}} We use $\infkdpckd$ as an infix operator  corresponding to applying term ${\mname{div\_SNo}}$.
\begin{theorem}\label{thm:jgogehpfdfoepgpfdgbgdhfgdh} ${\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{({\unicdcdadad n \in {\mname{omega}} .  {{p}~{n}}})}\unicdbdjdcd{{({\unicdcdadad n \in {\mname{omega}} .  {{p}~{({\prenc{{n}}})}}})}\unicdbdjdcd{\unicdcdadad x \in {\mname{int}} .  {{p}~{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2504c05a08587fe0873ed45685efc417625f0a904281d653d757d643896f9a70 \\
Pure Prop Address: & TMUEfaEhnmjqzhHDct4aN4nYsgTMNHexc5Z \\
Theory Prop Id: & a32835e1d8cbee7ebb59bff60b86d8d343a78a5a5b0a1a988fe8ae0752f9b0a9 \\
Theory Prop Address: & TMcPpJsLS8zQSHciyZymBNG61rHAnJoGbht
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dffhcgbhpfpgngfghgbgpfjgogeh} ${{\mname{omega}} \subseteq {\mname{int}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c213ff287d87049b1e6a47a232f87c366800922741a9eeadb1d3ac2fbadaf052 \\
Pure Prop Address: & TMbZKRVRA2o1bkPt9NF2BWMQLKKJzDaq77o \\
Theory Prop Id: & c3d99d24216bf364de569b6a9cd1de13b46ceb52421e78a517df07524d1308a5 \\
Theory Prop Address: & TMJyopLK6GuuCWqhP1XKASZh9iSwtdLiMs2
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:jgogehpfngjgogfhdhpfdfoepgpfpgngfghgbg} ${\unicdcdadad n \in {\mname{omega}} .  {{\prenc{{n}}} \in {\mname{int}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a66fb27a7b2af57722c6537d3983b55a12cc28475f1d8b8d9bdb1d857010e7af \\
Pure Prop Address: & TMHaC3837eh4SUn1jL2MVZSY2HYJVykA2q8 \\
Theory Prop Id: & 1b4d50647e615e454ab958cf206bb3f4b349286064ed07e205cde10ee49537b0 \\
Theory Prop Address: & TMYLKPRR9oEXo9N8Z2YBDanBt7x7VLvfePj
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:jgogehpfngjgogfhdhpfdfoepg} ${\unicdcdadad x \in {\mname{int}} .  {{\prenc{{x}}} \in {\mname{int}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 66d7a7b7f8768657be1ea35e52473cc5e1846e635153a280e3783a8275062773 \\
Pure Prop Address: & TMSXiWmTFpmbC8vtTafxAwFeYuNrgZJj4vW \\
Theory Prop Id: & daaadd58a6cfb975a0d9200d19c663526ca0a4951284ec16a5e11ae2e3fd7492 \\
Theory Prop Address: & TMV6sHDGtUc6XzQK9XLPBGFqCcMjQdPpFHz
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:jgogehpfbgegegpfdfoepgpfmgfgng} ${\unicdcdadad n \in {\mname{omega}} .  {\unicdcdadad m .  {{{\mname{nat\_p}}~{m}}\unicdbdjdcd{{{\prenc{{n}}}+{m}} \in {\mname{int}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6c976be5ae7c4eec61e1190f4b65a1cc39ebfb81542ae63578b69be42c01a06a \\
Pure Prop Address: & TMLvPoJ9aN37XYyR84rMi7xqr8n8tsuhtfM \\
Theory Prop Id: & 6fbb03f9ad1cac7ae41730f8a527dac94e9c83e918287c89f3787c4a3faac1a5 \\
Theory Prop Address: & TMPwH3A6N2tb87tNY5vbkqqydAuD5QR3EyF
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:jgogehpfbgegegpfdfoepg} ${\unicdcdadad x y \in {\mname{int}} .  {{{x}+{y}} \in {\mname{int}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 02609f82bf442d61fb8c6410818e7e4cbf8c67c9ea08bffcbf8a77c06b0f784a \\
Pure Prop Address: & TMFDVQawMT1qNmvZaWsWLAFRwNsZDvdEWd7 \\
Theory Prop Id: & a0aa5766d0cdd5590d613a2f72f025acfa458bc0cda8293f060de4d24de253c5 \\
Theory Prop Address: & TMag1esGeGDfnX5JdbJHct4N6L993dNSRki
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:jgogehpfngfhmgpfdfoepg} ${\unicdcdadad x y \in {\mname{int}} .  {{{x}*{y}} \in {\mname{int}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3f000c087708670f2a9497d0587231f65beb85593b7ebdaf7909fe5e7d4b8c27 \\
Pure Prop Address: & TMbhP3nSug9dzpwnhRm7VfyD7KAzF5WfghL \\
Theory Prop Id: & eefdfbfec3e66c305c9c9bc9c837268558851ac36fad12ee4b371f995afc9fd5 \\
Theory Prop Address: & TMbSE7nyU8X9azWYyGBkvTt8zRNBARSeYkH
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:egjgghjgegfgdhpfjgogeh}
We define $\mname{divides\_int}$ to be ${\uniddcgcg m n .  {{{{m} \in {\mname{int}}}\unicdcdcdhd{{n} \in {\mname{int}}}}\unicdcdcdhd{\unicdcdaddd k \in {\mname{int}} .  {{{m}*{k}}={n}}}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & af5f6e65098725dbbf9014383760c67fe7961b5486f9b4fa4d3289ed424f0c1b \\
Pure Object Address: & TMX9HJKhzi9AdyH6nXyJVdE4nKLJvyF9yNE \\
Theory Object Id: & 5fa28bc7b02e3ca96f01706f3aa25b382b44df0d71c91f7a7c68aea0935e9273 \\
Theory Object Address: & TMT9KA9dVEU4Cq2ursSu7Ep2iNaWdDT8u12
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:fgbhfhjgghpfjgogehpfngpgeg}
We define $\mname{equiv\_int\_mod}$ to be ${\uniddcgcg m k n .  {{{{{m} \in {\mname{int}}}\unicdcdcdhd{{k} \in {\mname{int}}}}\unicdcdcdhd{{n} \in {{\mname{omega}}\unicdcdbdgd{1}}}}\unicdcdcdhd{{{\mname{divides\_int}}~{({{m}+{\prenc{{k}}}})}}~{n}}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & aea710807b7901288ae25daacdd1e52348dfbe79c9a58fabb5e91c68e15a1581 \\
Pure Object Address: & TMHGhKGHRdgbdvTgpBF85dWTWtVw1PHGZZM \\
Theory Object Id: & a0bc48ffd10d4e9112ad9d98af3c6392a81e8ef3d019ce4b8ec86f77a5a49a7d \\
Theory Object Address: & TMWqX7aoEj8tNfrfptrVaGaaFkknJoVdiPo
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:dgpgahchjgngfgpfjgogeh}
We define $\mname{coprime\_int}$ to be ${\uniddcgcg a b .  {{{{a} \in {\mname{int}}}\unicdcdcdhd{{b} \in {\mname{int}}}}\unicdcdcdhd{\unicdcdadad x \in {{\mname{omega}}\unicdcdbdgd{1}} .  {{{{\mname{divides\_int}}~{x}}~{a}}\unicdbdjdcd{{{{\mname{divides\_int}}~{x}}~{b}}\unicdbdjdcd{{x}={1}}}}}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 48ca77268e35362c65956d7fd4145d07598e7a91e7ef4a1e6af043bc1202b37f \\
Pure Object Address: & TMRPbZHuWZc21V3zuKPZMGa165KDXe2rvuU \\
Theory Object Id: & 76fc5b183d6bea5474424b2b6616dbb8c29591fb620aa364408cc43266fa15ec \\
Theory Object Address: & TMXXHSe9mRiHSQ717podNGPQLawz24s1P1q
\end{tabular}
\end{center}

\end{definition}
\section{Packing Two Binary Operations, a Relation and Two Constants as a Set} % Int
\begin{definition}\label{def:ahbgdglgpfcgpfcgpfchpffgpffg}
$\mname{pack\_b\_b\_r\_e\_e}$ is the opaque object of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 8efb1973b4a9b292951aa9ca2922b7aa15d8db021bfada9c0f07fc9bb09b65fb \\
Pure Object Address: & TMFRc8E8tg6JtPF2W7a5hcH7dKma4rhhjpe \\
Theory Object Id: & a33bdffb44829c458189fcf9c1fc93f1abfb0007a9318b33fa18bab49932b69b \\
Theory Object Address: & TMUoWdaNvqJBEvaP7AZFqh425UCXiK49xiT
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfadpffgbh} ${\unicdcdadad S X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{S}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}\unicdbdjdcd{{X}={{S}~{0}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f5a2203f23e7dd04fe8ca0e3fb972d0098d3812a6f2a49e5016460d7db993bf5 \\
Pure Prop Address: & TMRGip4djr41vccrTDQXNBbxCWrX8VbHcqJ \\
Theory Prop Id: & 2c10ca5a1162991d07d3c553c21901d5abd7b71cb1ca8c024781096deed06657 \\
Theory Prop Address: & TMQjRgtSbPqLcPdbvF5gpcTyqr1VGnTtfgw
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfadpffgbhcd} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{X}={{{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}~{0}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7136b27641c7cc4e1a647a898e4ebf0836d4299f8da51ec9709fc246183a9280 \\
Pure Prop Address: & TMXU7ECTAkVR3MxsrCyo2KrDJ4eLHVp18Z2 \\
Theory Prop Id: & eb82f1296debb50aec472ad719325c4d415f74e68c24cbfeb6eb48e13b06fe27 \\
Theory Prop Address: & TMHkCAK4V2154ECdQyfXBNV3CMQ6datiFHq
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfbdpffgbh} ${\unicdcdadad S X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{S}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}\unicdbdjdcd{\unicdcdadad x y \in {X} .  {{{{f}~{x}}~{y}}={{{{\mname{decode\_b}}~{({{S}~{1}})}}~{x}}~{y}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ccc17a15fee3ff8e50e0a429d8fc3703e0d4322e680a51e8331926734edd0a78 \\
Pure Prop Address: & TMGv6tf3d83HWkjLGc4Bjfk2yL6bP1Y69Ea \\
Theory Prop Id: & b982a46d670799d817f2c72da1c2d0fb616888b041e52fcb1bdd4cd4f36e4a15 \\
Theory Prop Address: & TMPFdacX88nXYB1oHJX48p69cohuPHPDyHj
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfbdpffgbhcd} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {\unicdcdadad x y \in {X} .  {{{{f}~{x}}~{y}}={{{{\mname{decode\_b}}~{({{{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}~{1}})}}~{x}}~{y}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e738dbc94ff336d55dbd404f3523f553aed8269c125ce07c558108d936f8bf8e \\
Pure Prop Address: & TMczpSJ93XaFjvH4vijE4cc2RjpSvLCBnZ1 \\
Theory Prop Id: & 3048f770c8c092a3cc8c55bb78a7d82f19565aac3ff6652ef9d55e6c0cfb0962 \\
Theory Prop Address: & TMGxoR9fiBCzsbqU2uTbeELHpxWJM9ikYSB
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfcdpffgbh} ${\unicdcdadad S X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{S}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}\unicdbdjdcd{\unicdcdadad x y \in {X} .  {{{{g}~{x}}~{y}}={{{{\mname{decode\_b}}~{({{S}~{2}})}}~{x}}~{y}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a822390210aa0eeae11feeec870412a9d3f167449db2716772c8d85d36fde3ad \\
Pure Prop Address: & TMGuyMyeVJR3BDF1pJNw6KGpyy6D5ZNkVqb \\
Theory Prop Id: & 2d8185611c6998b74706354379f7e37a3983d2acac54f2a80803b05199954f8f \\
Theory Prop Address: & TMNWRvZ8TbLk6nt1L5aUcRmKEgigkF4bzbC
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfcdpffgbhcd} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {\unicdcdadad x y \in {X} .  {{{{g}~{x}}~{y}}={{{{\mname{decode\_b}}~{({{{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}~{2}})}}~{x}}~{y}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0904eb84a994cf26e71f4541979476936ad04090940f84ea7e3ae166191f386f \\
Pure Prop Address: & TMZgHEgS5WRZZnmn2WtWxSCUaL6S8KAKuEU \\
Theory Prop Id: & 514b7ec403837110f3c264073db0858ef3b66bcb89b034df3d99771f31f409f1 \\
Theory Prop Address: & TMPsaDHiTWNr1DquvQg49N7jxY2zPJAhBQD
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfddpffgbh} ${\unicdcdadad S X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{S}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}\unicdbdjdcd{\unicdcdadad x y \in {X} .  {{{{R}~{x}}~{y}}={{{{\mname{decode\_r}}~{({{S}~{3}})}}~{x}}~{y}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 20b01269edbb5f190d76d15ed12389468b65b4f3a49a6747fc5a644de89692c4 \\
Pure Prop Address: & TMZCKwwQH2ns9uknt6pfpXLnMeYPTdrkKsA \\
Theory Prop Id: & 543a595f6d03cb89cc51280f13fd9710e66da890237a5f792e79701620885eb7 \\
Theory Prop Address: & TMPLwektPtu4AjrXp4ghumX9jTbcGU7yeaD
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfddpffgbhcd} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {\unicdcdadad x y \in {X} .  {{{{R}~{x}}~{y}}={{{{\mname{decode\_r}}~{({{{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}~{3}})}}~{x}}~{y}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 84e5f9f9566add41b035a90f33b58ab881576661927e7112435c4cc63f5b9a82 \\
Pure Prop Address: & TMcKPp5Wtf1GqSMtPaoCwtuAz9G543UZXY7 \\
Theory Prop Id: & e6c65fe282c292809ca69c9ff83b219fff807abed4e4977ed7b6ca1bd889aeb5 \\
Theory Prop Address: & TMH676J4Qexg1jep9yQTFXu4vNi1TJ1qrZX
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfedpffgbh} ${\unicdcdadad S X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{S}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}\unicdbdjdcd{{c}={{S}~{4}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7c411fa1b904b5c366996dc15f10c49f1ca9b84a3441266671af3c8907575779 \\
Pure Prop Address: & TMSSAAEDYMUWREf5eqjEN4jmyCnfvyeobwE \\
Theory Prop Id: & 4e1693db4de27914c071907d430af0401ef3bd6ac24367ac8791b6c5c512091e \\
Theory Prop Address: & TMGzMNkeS7o5At8847MUjvdcHvigbTpaypB
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfedpffgbhcd} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{c}={{{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}~{4}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 79c12fb164791187a04849be1011349dce3e15aad6a3329faac347fc335089f5 \\
Pure Prop Address: & TMFqB5mPoMCs8LrcpMtffiyDeommry4W3Wf \\
Theory Prop Id: & 11df0febe6f1e3560b5041849da4dd2bdc05c8527c22c5617386567a92e57fa6 \\
Theory Prop Address: & TMct6V7NrbQ2y9kAcrjnEM2WVDGaQZrcgjK
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpffdpffgbh} ${\unicdcdadad S X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{S}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}\unicdbdjdcd{{d}={{S}~{5}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 94b3c3a63e1902ef3037b87eaa7fcbbae94338cd1163cceeed376f1eb8ec7ef1 \\
Pure Prop Address: & TMHxq3wxovqDHYziUjPoyA45c2sAhm2UNoq \\
Theory Prop Id: & cc915a2d36f6cc79c293654aab8fd3adb2808299e9b173a1c39bf3cdac0dfe5c \\
Theory Prop Address: & TMSYohXBC39TSSsVBLMnYazqxd2k9SYkjSg
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpffdpffgbhcd} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{d}={{{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}~{5}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b0d3a42291179839984699106279142ceeb8c26475897616c224628ed6d51873 \\
Pure Prop Address: & TMcWSqvc4wwoPFxLaHnaqahY7ZNdALEXxz1 \\
Theory Prop Id: & 29ae629ba4f8c9f4ea56fe01e58c7e5adc9a3f4032832a6256e4172bc1857db1 \\
Theory Prop Address: & TMSx1WE5sJ7RC3YQth9f6oafrR3W5T8cUAW
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpfjgogkg} ${\unicdcdadad X X' .  {\unicdcdadad f f' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g g' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R R' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c c' : {\iota} .  {\unicdcdadad d d' : {\iota} .  {{{{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X'}}~{f'}}~{g'}}~{R'}}~{c'}}~{d'}}}\unicdbdjdcd{{{{{{{X}={X'}}\unicdcdcdhd{({\unicdcdadad x y \in {X} .  {{{{f}~{x}}~{y}}={{{f'}~{x}}~{y}}}})}}\unicdcdcdhd{({\unicdcdadad x y \in {X} .  {{{{g}~{x}}~{y}}={{{g'}~{x}}~{y}}}})}}\unicdcdcdhd{({\unicdcdadad x y \in {X} .  {{{{R}~{x}}~{y}}={{{R'}~{x}}~{y}}}})}}\unicdcdcdhd{{c}={c'}}}\unicdcdcdhd{{d}={d'}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 41bdc7cea27f40d41bd47e10070a9221fac56ee80059ed6ef1145c640f0a0696 \\
Pure Prop Address: & TMZzLqefgu7rLgWmGbXgEsJw9KyJ9HfnEfv \\
Theory Prop Id: & 030a267e64cc2ab4e90b427a38635de36741b1d1f5f749729223836ad895b355 \\
Theory Prop Address: & TMFdLKT9s9pybGa29WS8rRsZtaZEB878Tqc
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfcgpfcgpfchpffgpffgpffgiheh} ${\unicdcdadad X .  {\unicdcdadad f f' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g g' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R R' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c .  {\unicdcdadad d .  {{({\unicdcdadad x y \in {X} .  {{{{f}~{x}}~{y}}={{{f'}~{x}}~{y}}}})}\unicdbdjdcd{{({\unicdcdadad x y \in {X} .  {{{{g}~{x}}~{y}}={{{g'}~{x}}~{y}}}})}\unicdbdjdcd{{({\unicdcdadad x y \in {X} .  {{{{R}~{x}}~{y}}\unicdbdjded{{{R'}~{x}}~{y}}}})}\unicdbdjdcd{{{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f'}}~{g'}}~{R'}}~{c}}~{d}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 65aff8c9424ee01cd07c15ae87a00a524f02d18e2ec143a5bba84026587bf7f3 \\
Pure Prop Address: & TMcK19CLntYiEUmEtDidiA4HpkWUA29DCY8 \\
Theory Prop Id: & 37b535ddadbc59098fcbd9eea3333e24e8d8ba92e8fa6adac43561bcb663a641 \\
Theory Prop Address: & TMSrpevWvuK3PWAB7qQDUur4T35U2y5PcFh
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dhehchfhdgehpfcgpfcgpfchpffgpffg}
We define $\mname{struct\_b\_b\_r\_e\_e}$ to be ${\uniddcgcg S .  {\unicdcdadad q : {{\iota}\unicdbdjdcd{o}} .  {{({\unicdcdadad X : {\iota} .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {{({\unicdcdadad x y \in {X} .  {{{{f}~{x}}~{y}} \in {X}}})}\unicdbdjdcd{\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {{({\unicdcdadad x y \in {X} .  {{{{g}~{x}}~{y}} \in {X}}})}\unicdbdjdcd{\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {{{c} \in {X}}\unicdbdjdcd{\unicdcdadad d : {\iota} .  {{{d} \in {X}}\unicdbdjdcd{{q}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}})}}}}}}}}}}}})}\unicdbdjdcd{{q}~{S}}}}}$ of type ${{\iota}\unicdbdjdcd{o}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & d542f60aebdbe4e018abe75d8586699fd6db6951a7fdc2bc884bfb42c0d77a22 \\
Pure Object Address: & TMXf978YVAAD5ZQex2ymHckFjvhRj8VqJG9 \\
Theory Object Id: & 4b668479c87844857651881d76dfae59eab34bb3676e61f76f6dd64b6060cf5d \\
Theory Object Address: & TMKS3b6mms1vofrtcap5oMtL9Kdzd24UwRB
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:ahbgdglgpfdhehchfhdgehpfcgpfcgpfchpffgpffgpfje} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {{({\unicdcdadad x y \in {X} .  {{{{f}~{x}}~{y}} \in {X}}})}\unicdbdjdcd{\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {{({\unicdcdadad x y \in {X} .  {{{{g}~{x}}~{y}} \in {X}}})}\unicdbdjdcd{\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {{{c} \in {X}}\unicdbdjdcd{\unicdcdadad d : {\iota} .  {{{d} \in {X}}\unicdbdjdcd{{\mname{struct\_b\_b\_r\_e\_e}}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}})}}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 837b01cc10cf77ecfed9c05cb135bfee5d5785cb3eeb155cd67420a1e8117277 \\
Pure Prop Address: & TMdKBXs47VAuzq32DPCspAGpMJxWtP7HZgn \\
Theory Prop Id: & 4e1d4bac0d2464119ef3ae868cd30477053be4e52d11f44a4826bb32b543f78a \\
Theory Prop Address: & TMRfKTdq9BR53wNTCPty6ckkNQRPzNXtwXR
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfdhehchfhdgehpfcgpfcgpfchpffgpffgpffebd} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{\mname{struct\_b\_b\_r\_e\_e}}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}})}}\unicdbdjdcd{\unicdcdadad x y \in {X} .  {{{{f}~{x}}~{y}} \in {X}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 33666944863c7073c5d24b456627c47939a6c292049de068c2dd0a2696024881 \\
Pure Prop Address: & TMbqCqWUV4Qr8yRDenPHW1SCXnMHwtzsEqR \\
Theory Prop Id: & bce7cd0660e462218e87e9b6e9756ae8795d7e47a2034af9f01a233591cd4516 \\
Theory Prop Address: & TMTcgTJU3oXQrtzipjvwHThz1WBYVpT6Tr8
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfdhehchfhdgehpfcgpfcgpfchpffgpffgpffecd} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{\mname{struct\_b\_b\_r\_e\_e}}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}})}}\unicdbdjdcd{\unicdcdadad x y \in {X} .  {{{{g}~{x}}~{y}} \in {X}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8d7f819bc88f04c8fb38f9c45656b24be62467c1111be1ab6669a362d2c42bd9 \\
Pure Prop Address: & TMFq66jAxt6ToaMefzyqpZmo2jxWBS1L5KS \\
Theory Prop Id: & a642313376049cb0e12a05cfd42eb6fe35a06fa7ac25ab566e3a668a2f094fa5 \\
Theory Prop Address: & TMKRXGcsc5vFBxDbQ738RFuErE5UAsvbXrX
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfdhehchfhdgehpfcgpfcgpfchpffgpffgpffeed} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{\mname{struct\_b\_b\_r\_e\_e}}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}})}}\unicdbdjdcd{{c} \in {X}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fe2c180cc0cf649c36df0e3a2f9dabf02603a9d163927935d608712fe908c0a0 \\
Pure Prop Address: & TMKkUcfVuzqdAAqixhrJdaDtwAiwP588iFb \\
Theory Prop Id: & 4343532cd958d270bb174fb9ef0d486e82bcb7068472964cad87b3aa59268c36 \\
Theory Prop Address: & TMVkwckBqwmZ3ptsoBAtwgWpGJnBf1HEKJX
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:ahbgdglgpfdhehchfhdgehpfcgpfcgpfchpffgpffgpffefd} ${\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{{\mname{struct\_b\_b\_r\_e\_e}}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}})}}\unicdbdjdcd{{d} \in {X}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1eaba4d18e5ef9249157efebe811892c11f22eaccbdfaabe7207d0eed3610753 \\
Pure Prop Address: & TMN8VR6KEzNcbMfJUKCFWE9VrqsSAghpm2a \\
Theory Prop Id: & d7fee779a369a0d4bd81cd98ec4b107047599cd8daebdc70621edcfba45ab303 \\
Theory Prop Address: & TMRKtcvjctfdAsskw9TbK95G7FLTa6F6w99
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:dhehchfhdgehpfcgpfcgpfchpffgpffgpffgehbg} ${\unicdcdadad S .  {{{\mname{struct\_b\_b\_r\_e\_e}}~{S}}\unicdbdjdcd{{S}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{({{S}~{0}})}}~{({{\mname{decode\_b}}~{({{S}~{1}})}})}}~{({{\mname{decode\_b}}~{({{S}~{2}})}})}}~{({{\mname{decode\_r}}~{({{S}~{3}})}})}}~{({{S}~{4}})}}~{({{S}~{5}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 08c6efad770174c53c57bf1abbcff7a0b238338e1a0a913fdb1e12ac817545eb \\
Pure Prop Address: & TMGztHkuyt42zcYRkYwSMfmeqnNcm8EkxuB \\
Theory Prop Id: & 3a4c426298d981750f4f7847d710686bde28aba7b880e1e16dc568078e493ab5 \\
Theory Prop Address: & TMdFRkJgJNNXSyM8pAQoyEqPBnQPHMehaWr
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:fhogahbgdglgpfcgpfcgpfchpffgpffgpfjg}
$\mname{unpack\_b\_b\_r\_e\_e\_i}$ is the opaque object of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}}}}})}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 89fb5e380d96cc4a16ceba7825bfbc02dfd37f2e63dd703009885c4bf3794d07 \\
Pure Object Address: & TMV6sraHemVG7Q2TgN6oCsJmkmebyJuYVNd \\
Theory Object Id: & 76f5f924237f946f8a80d7200b117ad49d05a912a831233242659e6f501bf978 \\
Theory Object Address: & TMRP69PDPah6ZkTeGTDbp6iVPip8vSRTsB8
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:fhogahbgdglgpfcgpfcgpfchpffgpffgpfjgpffgbh} ${\unicdcdadad Phi : {{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}}}}} .  {\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{({\unicdcdadad f' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {{({\unicdcdadad x y \in {X} .  {{{{f}~{x}}~{y}}={{{f'}~{x}}~{y}}}})}\unicdbdjdcd{\unicdcdadad g' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {{({\unicdcdadad x y \in {X} .  {{{{g}~{x}}~{y}}={{{g'}~{x}}~{y}}}})}\unicdbdjdcd{\unicdcdadad R' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {{({\unicdcdadad x y \in {X} .  {{{{R}~{x}}~{y}}\unicdbdjded{{{R'}~{x}}~{y}}}})}\unicdbdjdcd{{{{{{{{Phi}~{X}}~{f'}}~{g'}}~{R'}}~{c}}~{d}}={{{{{{{Phi}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}}}}}}})}\unicdbdjdcd{{{{\mname{unpack\_b\_b\_r\_e\_e\_i}}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}})}}~{Phi}}={{{{{{{Phi}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b58f007e91d43a2c7a09dc0b595961936ec7da158803e7b98c8e478584d206ef \\
Pure Prop Address: & TMPqkpAFSVWhtBYDqJsL7JfE4qYGQvWKVkp \\
Theory Prop Id: & 9a7986d0258cb1c3270b60b6b454888095d2aa2a2a53d845b4756902a36876f7 \\
Theory Prop Address: & TMQfsPU4J48cc3XAFwauAcYbj5YSbwJVxJw
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:fhogahbgdglgpfcgpfcgpfchpffgpffgpfpg}
$\mname{unpack\_b\_b\_r\_e\_e\_o}$ is the opaque object of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}}}}})}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & b3a2fc60275daf51e6cbe3161abaeed96cb2fc4e1d5fe26b5e3e58d0eb93c477 \\
Pure Object Address: & TMP75qXVTx8SpwHV1y6wefKnbFqVV7VPCV7 \\
Theory Object Id: & fc0d55fe8a09aad6902204ac4550a6671cc57b3b29e4c36503dea3e8cc818737 \\
Theory Object Address: & TMQVwZxiv8LXLzFAAtQUdESpWctYhNPKuLQ
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:fhogahbgdglgpfcgpfcgpfchpffgpffgpfpgpffgbh} ${\unicdcdadad Phi : {{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}}}}} .  {\unicdcdadad X .  {\unicdcdadad f : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad g : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad R : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad c : {\iota} .  {\unicdcdadad d : {\iota} .  {{({\unicdcdadad f' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {{({\unicdcdadad x y \in {X} .  {{{{f}~{x}}~{y}}={{{f'}~{x}}~{y}}}})}\unicdbdjdcd{\unicdcdadad g' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {{({\unicdcdadad x y \in {X} .  {{{{g}~{x}}~{y}}={{{g'}~{x}}~{y}}}})}\unicdbdjdcd{\unicdcdadad R' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {{({\unicdcdadad x y \in {X} .  {{{{R}~{x}}~{y}}\unicdbdjded{{{R'}~{x}}~{y}}}})}\unicdbdjdcd{{{{{{{{Phi}~{X}}~{f'}}~{g'}}~{R'}}~{c}}~{d}}={{{{{{{Phi}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}}}}}}})}\unicdbdjdcd{{{{\mname{unpack\_b\_b\_r\_e\_e\_o}}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}})}}~{Phi}}={{{{{{{Phi}~{X}}~{f}}~{g}}~{R}}~{c}}~{d}}}}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b4ebc419abea98259a9b2fc6ee430a9eb6b66901e01778798caef327518ab73c \\
Pure Prop Address: & TMWy9NbJuezT2C2YvSVs4gNoW6RxDCu36cJ \\
Theory Prop Id: & 308ccf9882d7cb03559534a600b368145e2c7062734d3ea9574a2b00c5a6bbd1 \\
Theory Prop Address: & TMdarJDcWV77zy8PF3SrMpz5YVBUqiWCp7G
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:pechegfgchfgeggejgfgmgegdfehchfhdgeh}
$\mname{OrderedFieldStruct}$ is the opaque object of type ${{\iota}\unicdbdjdcd{o}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 98aa98f64160bebd5622706908b639f9fdfe4056f2678ed69e5b099b8dd7b888 \\
Pure Object Address: & TMEzWdp7RFmZMEiX1EsMMaPEcXjJJfCtETX \\
Theory Object Id: & 1de6a120bae49aa4021a9a6626425398f0c8d34c17a449aca28902eba9cfe754 \\
Theory Object Address: & TMWMkhjM35KDgsppiv3Up8VXcf7pNB4BC6Q
\end{tabular}
\end{center}

\end{definition}
\section{explicit\_OrderedField\_RepIndep2}\label{fgihahmgjgdgjgehpfpechegfgchfgeggejgfgmgegpfcffgahjeogegfgahcd}Let $ R : {\iota}$ be given.
Let $ zero one : {\iota}$ be given.
Let $ plus mult : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ be given.
Let $ leq : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ be given.
{\bf{Notation.}} We use $\inflc$ as a right associative infix operator  corresponding to applying term ${plus}$.
{\bf{Notation.}} We use $\infkc$ as a right associative infix operator  corresponding to applying term ${mult}$.
Let $ plus' mult' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ be given.
Let $ leq' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ be given.
{\bf{Notation.}} We use $\infkdlckd$ as a right associative infix operator  corresponding to applying term ${plus'}$.
{\bf{Notation.}} We use $\infkdkckd$ as a right associative infix operator  corresponding to applying term ${mult'}$.
Assume the following.
\begin{equation}
{\unicdcdadad a b \in {R} .  {{{a}+{b}}={{a}\uniadadcdcg{b}}}}\label{hyp:ieahahhc}
\end{equation}
Assume the following.
\begin{equation}
{\unicdcdadad a b \in {R} .  {{{a}*{b}}={{a}\unicdbgcdgg{b}}}}\label{hyp:iengnghc}
\end{equation}
Assume the following.
\begin{equation}
{\unicdcdadad a b \in {R} .  {{{{leq}~{a}}~{b}}\unicdbdjded{{{leq'}~{a}}~{b}}}}\label{hyp:iemgmghc}
\end{equation}
\begin{theorem}\label{thm:fgihahmgjgdgjgehpfpechegfgchfgeggejgfgmgegpfchfgahjgogegfgah} ${{{{{{{{\mname{explicit\_OrderedField}}~{R}}~{zero}}~{one}}~{plus}}~{mult}}~{leq}}\unicdbdjded{{{{{{{\mname{explicit\_OrderedField}}~{R}}~{zero}}~{one}}~{plus'}}~{mult'}}~{leq'}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & aa6280ff9ecfb388daf95ebbc769d81f90cc6039e5299709d31d625891887057 \\
Pure Prop Address: & TMbsFvtTvqkK3SszQVqXzQ1UJEsUnVYiMZL \\
Theory Prop Id: & be25ce3a19ac950fdaa6ea42fe6fcb818e428c8dc8aae4b4fbe47fa06d08e029 \\
Theory Prop Address: & TMdEcCwhu3D2kVgdgGVo1Cmg5W1sXuLX3HR
\end{tabular}
\end{center}

\end{theorem}
\section{Unpacking Ordered Fields} % explicit_OrderedField_RepIndep2
\begin{theorem}\label{thm:pechegfgchfgeggejgfgmgegdfehchfhdgehpffhogahbgdglgpffgbh} ${\unicdcdadad R .  {\unicdcdadad plus mult : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad leq : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad zero one .  {{{{\mname{unpack\_b\_b\_r\_e\_e\_o}}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{R}}~{plus}}~{mult}}~{leq}}~{zero}}~{one}})}}~{({\uniddcgcg R plus mult leq zero one .  {{{{{{{\mname{explicit\_OrderedField}}~{R}}~{zero}}~{one}}~{plus}}~{mult}}~{leq}}})}}={{{{{{{\mname{explicit\_OrderedField}}~{R}}~{zero}}~{one}}~{plus}}~{mult}}~{leq}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0f64cee1eaa56826e0a1df23c3c822f54eeeaf346766143d43f9cd8aaa5bcc05 \\
Pure Prop Address: & TMcaQWXF265MqyQdZxoLdbgCWynQwDq1QFz \\
Theory Prop Id: & 6fe0fe98ce1675b6b7d6602c3080c16c71b29a61d4b004d43b5855aad4d9a43f \\
Theory Prop Address: & TMXMSqH6fsZZwHid44hbcR3wDdTqU7Dm9HQ
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgeh}
We define $\mname{RealsStruct}$ to be ${\uniddcgcg R .  {{{\mname{struct\_b\_b\_r\_e\_e}}~{R}}\unicdcdcdhd{{{\mname{unpack\_b\_b\_r\_e\_e\_o}}~{R}}~{({\uniddcgcg R plus mult leq zero one .  {{{{{{{\mname{explicit\_Reals}}~{R}}~{zero}}~{one}}~{plus}}~{mult}}~{leq}}})}}}}$ of type ${{\iota}\unicdbdjdcd{o}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 9971dcf75f0995efe4245a98bcdf103e4713261d35432146405052f36f8234bf \\
Pure Object Address: & TMJjjbmFf1HjinfhpdrL5vryqPNmEPLfGtv \\
Theory Object Id: & 3d1a749dfea9bb7211b105adb25ae072c40841648e843f5cbc5483cd7b1257cd \\
Theory Object Address: & TMQydd7NRsjg2p9tCMw9pgNmAnEKZZD616o
\end{tabular}
\end{center}

\end{definition}
\section{explicit\_Reals\_RepIndep2}\label{fgihahmgjgdgjgehpfcffgbgmgdhpfcffgahjeogegfgahcd}Let $ R : {\iota}$ be given.
Let $ zero one : {\iota}$ be given.
Let $ plus mult : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ be given.
Let $ leq : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ be given.
{\bf{Notation.}} We use $\inflc$ as a right associative infix operator  corresponding to applying term ${plus}$.
{\bf{Notation.}} We use $\infkc$ as a right associative infix operator  corresponding to applying term ${mult}$.
Let $ plus' mult' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ be given.
Let $ leq' : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ be given.
{\bf{Notation.}} We use $\infkdlckd$ as a right associative infix operator  corresponding to applying term ${plus'}$.
{\bf{Notation.}} We use $\infkdkckd$ as a right associative infix operator  corresponding to applying term ${mult'}$.
Assume the following.
\begin{equation}
{\unicdcdadad a b \in {R} .  {{{a}+{b}}={{a}\uniadadcdcg{b}}}}\label{hyp:ieahahhc}
\end{equation}
Assume the following.
\begin{equation}
{\unicdcdadad a b \in {R} .  {{{a}*{b}}={{a}\unicdbgcdgg{b}}}}\label{hyp:iengnghc}
\end{equation}
Assume the following.
\begin{equation}
{\unicdcdadad a b \in {R} .  {{{{leq}~{a}}~{b}}\unicdbdjded{{{leq'}~{a}}~{b}}}}\label{hyp:iemgmghc}
\end{equation}
\begin{theorem}\label{thm:fgihahmgjgdgjgehpfcffgbgmgdhpfchfgahjgogegfgah} ${{{{{{{{\mname{explicit\_Reals}}~{R}}~{zero}}~{one}}~{plus}}~{mult}}~{leq}}\unicdbdjded{{{{{{{\mname{explicit\_Reals}}~{R}}~{zero}}~{one}}~{plus'}}~{mult'}}~{leq'}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e6777aed3a48a1eed98a6aa68ec40260a6fd7f77f99e3b2dfc8fd4213fbf51d5 \\
Pure Prop Address: & TMHRoUeAQwKBzV5DNUEnEtSGLyRQBCWi2oz \\
Theory Prop Id: & e7a9656809ed03974b3a3e7265ecbe46e51f37029b92e73c457801fb7c611216 \\
Theory Prop Address: & TMav3v4dLGD2kzFGsBAHjnvy2SFfYuJuB5m
\end{tabular}
\end{center}

\end{theorem}
\section{Unpacking Explicit Reals} % explicit_Reals_RepIndep2
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpffhogahbgdglgpffgbh} ${\unicdcdadad R .  {\unicdcdadad plus mult : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {\unicdcdadad leq : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}} .  {\unicdcdadad zero one .  {{{{\mname{unpack\_b\_b\_r\_e\_e\_o}}~{({{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{R}}~{plus}}~{mult}}~{leq}}~{zero}}~{one}})}}~{({\uniddcgcg R plus mult leq zero one .  {{{{{{{\mname{explicit\_Reals}}~{R}}~{zero}}~{one}}~{plus}}~{mult}}~{leq}}})}}={{{{{{{\mname{explicit\_Reals}}~{R}}~{zero}}~{one}}~{plus}}~{mult}}~{leq}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d753a2caecebad53fe06dc1e4aa5fc9c806b9df39093b6b31f3dfd7d2790d31f \\
Pure Prop Address: & TMKP2fsixYdw8yYkEFeLRmmhRgMFNqBiCkE \\
Theory Prop Id: & 1e4afd0f43b0a314315c5e7b2a4354fc8792a577d40f5ddbfe8054c2fd4bf78c \\
Theory Prop Address: & TMHSDGonYJvmsYAvpMXetztjTkxJWrGWgqS
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfdgbgchchjgfgch}
We define $\mname{RealsStruct\_carrier}$ to be ${\uniddcgcg Rs .  {{Rs}~{0}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & f0bb69e74123475b6ecce64430b539b64548b0b148267ea6c512e65545a391a1 \\
Pure Object Address: & TMY8sbcPiVs9VL311KPHpbm67U6sLM9JvF7 \\
Theory Object Id: & 9211e41eaf81bb8c152618cbe35230be60d736fcd5684c0b7bdc5ccc1ff70181 \\
Theory Object Address: & TMZQE6tPrFvJMaBcLrcMn85ZgscSeFbm4pT
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfahmgfhdh}
We define $\mname{RealsStruct\_plus}$ to be ${\uniddcgcg Rs .  {{\mname{decode\_b}}~{({{Rs}~{1}})}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 6968cd9ba47369f21e6e4d576648c6317ed74b303c8a2ac9e3d5e8bc91a68d9d \\
Pure Object Address: & TMYTM1QQGqgVyRM9unR8CLL2scwRHYvV3c3 \\
Theory Object Id: & ed06aa34b4112f8507370b46c1c5c7de7f2f9af4f24b831d45893ad3066e994e \\
Theory Object Address: & TMRiY7KU6BpU28KD19iwgBDLsQ5XyfpcbYL
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfngfhmgeh}
We define $\mname{RealsStruct\_mult}$ to be ${\uniddcgcg Rs .  {{\mname{decode\_b}}~{({{Rs}~{2}})}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 4b4b94d59128dfa2798a2e72c886a5edf4dbd4d9b9693c1c6dea5a401c53aec4 \\
Pure Object Address: & TMQSTL5b71ddbjnkjnGXfRZ61uXWmMGwJHD \\
Theory Object Id: & 7eb39ed695c795e4451a9252d011d377d0f4bdf0bfe6193f7a31ceb58f687a40 \\
Theory Object Address: & TMaJmzoi8UnDNTAXrAvZxYbTeEdKV3viygg
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfmgfgbh}
We define $\mname{RealsStruct\_leq}$ to be ${\uniddcgcg Rs .  {{\mname{decode\_r}}~{({{Rs}~{3}})}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & e59503fd8967d408619422bdebda4be08c9654014309b60880a3d7acf647e2b4 \\
Pure Object Address: & TMVpPCt6DnVyyzfVCfyhFErhhmLuHJWfLF5 \\
Theory Object Id: & e66070fe509c0522857fd1dec315953e7466e1fff9feba36b2b4e9f1f60f7ede \\
Theory Object Address: & TMGzX1JqbMSdTKAjW1hbwGwKwhVDCcRKEhA
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfkhfgchpg}
We define $\mname{RealsStruct\_zero}$ to be ${\uniddcgcg Rs .  {{Rs}~{4}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 57e734ae32adddc1f13e75e7bab22241e5d2c12955ae233ee90f53818028312a \\
Pure Object Address: & TMV8UDBZRqv4jocV6zt17aP89PcYugzQkoU \\
Theory Object Id: & b2fa78f7f58b95f4bc4f46ac268c3a959090897e56338c64bd84a73bc5fc2551 \\
Theory Object Address: & TMWFb9uZk2JEesS9quTe9a6Ww4ND9VcbBJA
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfpgogfg}
We define $\mname{RealsStruct\_one}$ to be ${\uniddcgcg Rs .  {{Rs}~{5}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 7aa92281bc38360837d18b9ec38ff8359d55a8c6ef4349c5777aab707e79589b \\
Pure Object Address: & TMG8fYn75tfBRqBJzcYzTPSyLmxrzMatNuC \\
Theory Object Id: & e902384fa204aa548a7192aca2937826cba222865eef88ccb0f0dd747aff3878 \\
Theory Object Address: & TMF5LB2qHkXJqUPWHWovgb3u9mde2rLtjtB
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:gejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgeh}
$\mname{Field\_of\_RealsStruct}$ is the opaque object of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & e1df2c6881a945043093f666186fa5159d4b31d68340b37bd2be81d10ba28060 \\
Pure Object Address: & TMYju5xqA4jiDon5Vrw64253Ybc4eYYipFW \\
Theory Object Id: & ef27c4f9f215b0e5024d128458fefa23faae900a3a796c7655191a5180aa8fd2 \\
Theory Object Address: & TMS4LA1a2HRZdid71r9n3J7xvnqcWXoPRYU
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:gejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgehpfad} ${\unicdcdadad Rs .  {{{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{0}}={{\mname{RealsStruct\_carrier}}~{Rs}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e3591bbc5c24313d727f99859f33bddaa24ed93f5e16fde0a5a27460f4deaede \\
Pure Prop Address: & TMGpDdtdmCUs1jrGPV3mZk171Ru5G3eeAgA \\
Theory Prop Id: & cb1fd7ac2423a50aaf869926ca1d1306d77aef6c4d3fa9f561447ef07b391cb8 \\
Theory Prop Address: & TMNmZN2BcVU3Bvv3THemmMQphYBi7VZEiK4
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:gejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgehpfbd} ${\unicdcdadad Rs .  {\unicdcdadad x y \in {{\mname{RealsStruct\_carrier}}~{Rs}} .  {{{{{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{1}}~{x}}~{y}}={{{{\mname{RealsStruct\_plus}}~{Rs}}~{x}}~{y}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a29ae8db462729b7f524ef65c8169e19aa80b02b813f1538b50a4ea71cac00e4 \\
Pure Prop Address: & TMH9MkMZY6Nfy8CtfBXnKbGTTMMDYtF8dhg \\
Theory Prop Id: & f4fd5c9274754011767f3f3a1963005a161b0e3450a8605917bb74939f42093a \\
Theory Prop Address: & TMHPEQeFskKiibC12Eg5RfK8N4RmZgBPc6M
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:gejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgehpfcd} ${\unicdcdadad Rs .  {\unicdcdadad x y \in {{\mname{RealsStruct\_carrier}}~{Rs}} .  {{{{{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{2}}~{x}}~{y}}={{{{\mname{RealsStruct\_mult}}~{Rs}}~{x}}~{y}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 144feb0ed55ae0791a3276243986374c09700d3d3d141752481d9130cc01a231 \\
Pure Prop Address: & TMQiqH6UGKCC2AY2aZmhwjBD1uigXrL9U5A \\
Theory Prop Id: & b9ac92481e26e620bc1d7f8daba3ccb4dcffbac3703a966f7f6175c0996df7a2 \\
Theory Prop Address: & TMHw73bPqVnCDmcBgyog1NWbZ2kKZNjpM2F
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:gejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgehpfdd} ${\unicdcdadad Rs .  {{{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{3}}={{\mname{RealsStruct\_zero}}~{Rs}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3a0c651ac07600c96cb2b4b797025bc57c3f63713cb95551e7ff7570600dd9a6 \\
Pure Prop Address: & TMP2bK3Zy9P941RMigpsJeoKpTUvZdbu6jt \\
Theory Prop Id: & 39ddb5b76ae60cb54984057402ef494c4fa1400d58d80b904f425a9a5e3609e3 \\
Theory Prop Address: & TMLDSxVyVzr1FPB6KncZtioj1PDrXC6M8SJ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:gejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgehpfed} ${\unicdcdadad Rs .  {{{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{4}}={{\mname{RealsStruct\_one}}~{Rs}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f8ef8a9601b742e71cf054c41b20f6ec0dee405cf311308f2196ca2ae8742478 \\
Pure Prop Address: & TMHKLv75keSDVRMA8BVfAdqxSSSWoVHCjjK \\
Theory Prop Id: & 79ba04006dd069d9bc477280db9ba7c3bf231bcc112d5aaa271be90cf51e8d30 \\
Theory Prop Address: & TMJTgymdjGtaV8G45xSwQDuSpRLtCjKY7LG
\end{tabular}
\end{center}

\end{theorem}
\section{RealsStruct}\label{cffgbgmgdhdfehchfhdgeh}Let $ Rs : {\iota}$ be given.
Assume the following.
\begin{equation}
{{\mname{RealsStruct}}~{Rs}}\label{hyp:iecfdh}
\end{equation}
Let $R : {\iota}$ be ${{\mname{RealsStruct\_carrier}}~{Rs}}$.
Let $zero : {\iota}$ be ${{\mname{RealsStruct\_zero}}~{Rs}}$.
Let $one : {\iota}$ be ${{\mname{RealsStruct\_one}}~{Rs}}$.
{\bf{Notation.}} We use $\inflc$ as a right associative infix operator  corresponding to applying term ${{\mname{RealsStruct\_plus}}~{Rs}}$.
{\bf{Notation.}} We use $\infkc$ as a right associative infix operator  corresponding to applying term ${{\mname{RealsStruct\_mult}}~{Rs}}$.
{\bf{Notation.}} We use $\infmdnd$ as an infix operator  corresponding to applying term ${{\mname{RealsStruct\_leq}}~{Rs}}$.
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpffgehbg} ${{Rs}={{{{{{{\mname{pack\_b\_b\_r\_e\_e}}~{R}}~{({{\mname{RealsStruct\_plus}}~{Rs}})}}~{({{\mname{RealsStruct\_mult}}~{Rs}})}}~{({{\mname{RealsStruct\_leq}}~{Rs}})}}~{zero}}~{one}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 17af86f27285e29398960ca5e9e3f6f45b69b7a9bdf827de09c0c669b7c75c6c \\
Pure Prop Address: & TMWmf9GomThUEUvw4iwAFWDwRvBCKuKUJ81 \\
Theory Prop Id: & f1cea2ca03fd17235452510cf22a8e1709b9373fee61b9d92e62557ba14e4287 \\
Theory Prop Address: & TMbTji4uUoLpyNMKCgYne4Fdo2wyrpD2a2z
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpffgihahmgjgdgjgehpfcffgbgmgdh} ${{{{{{{\mname{explicit\_Reals}}~{R}}~{zero}}~{one}}~{({{\mname{RealsStruct\_plus}}~{Rs}})}}~{({{\mname{RealsStruct\_mult}}~{Rs}})}}~{({{\mname{RealsStruct\_leq}}~{Rs}})}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d3877b9d93ebdb9c81bfb64ef324b904ac578f720458bd89b59d747bab8321ec \\
Pure Prop Address: & TMbsdad9BHngHtYG5Eu56f41hLLyMhQ4wz9 \\
Theory Prop Id: & e96bfd03b3e326ebe235a70e085a6663531d7408960525dca468e6023497d823 \\
Theory Prop Address: & TMMM29TxtZ5QApu738fkTwRqJ5AxhqnNU1q
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:gejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgehpfjgdhpfdecfjgoghgpfhhjgehigpfjgeg} ${{\mname{CRing\_with\_id}}~{({{\mname{Field\_of\_RealsStruct}}~{Rs}})}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 30ac8336eb76436c0b3b234f78de119af4f8aae671ed6946516746919bd40b57 \\
Pure Prop Address: & TMRwfFMfrGHNJdjEebj6AWUAsps8wxbBciR \\
Theory Prop Id: & 8d5700aa263b7f06933873e9cd11f836e00e0138000b4d39f89edcf89a4b5b69 \\
Theory Prop Address: & TMYnS1DaswnJWAykA7gfcHRYUno5P7ZH1xt
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfmgeh}
We define $\mname{RealsStruct\_lt}$ to be ${\uniddcgcg x y .  {{{x}\unicdcdgded{y}}\unicdcdcdhd{{x}\unicdcdgdad{y}}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 3a46d8baa691ebb59d2d6c008a346206a39f4baae68437520106f9673f41afc6 \\
Pure Object Address: & TMZU8qJtXFES85dykXqXncY6GKaYJb1UkwA \\
Theory Object Id: & 1fb825962c4e3c207604d053fa6fdb0b78c4215dab034f2c783d256d7e08a3f3 \\
Theory Object Address: & TMcanZ7iZKUNsAhX6fjudSebaubcUhfXhth
\end{tabular}
\end{center}

\end{definition}
{\bf{Notation.}} We use $\infmd$ as an infix operator  corresponding to applying term ${\mname{RealsStruct\_lt}}$.
\begin{theorem}\label{thm:fgihahmgjgdgjgehpfgejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgeh} ${{{{{{\mname{explicit\_Field}}~{R}}~{zero}}~{one}}~{({{\mname{RealsStruct\_plus}}~{Rs}})}}~{({{\mname{RealsStruct\_mult}}~{Rs}})}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d59c3f2e6f4d6724d3005e0bee526d84560fc597c159862117cd50437503db9a \\
Pure Prop Address: & TMYsjqWb3PES4DyoFitce9WwQU2Rt5WHX1i \\
Theory Prop Id: & 744fcb4f7b03a77f3f942ae46ec1dc0b6e66ac611a343175773867714b070aeb \\
Theory Prop Address: & TMairfquc4KvRjg7Xc3pANWUih3uqyq8R5j
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:fgihahmgjgdgjgehpfpechegfgchfgeggejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgeh} ${{{{{{{\mname{explicit\_OrderedField}}~{R}}~{zero}}~{one}}~{({{\mname{RealsStruct\_plus}}~{Rs}})}}~{({{\mname{RealsStruct\_mult}}~{Rs}})}}~{({{\mname{RealsStruct\_leq}}~{Rs}})}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 70df0d2f2fe64ff20992d794cafe8f45b02a15058699156c2ded116e9b0962e6 \\
Pure Prop Address: & TMUxQacVa4YowWpPGUq5zVNjPj5dQafMF9G \\
Theory Prop Id: & afda6089023853c8239e43e59fcb8eb5e74e269802cfec8e72ac2071ca787656 \\
Theory Prop Address: & TMUSiQQFDe6Uiw3xmogsw37fkZT92vjPKrA
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpechegfgchfgeggejgfgmgeg} ${{\mname{OrderedFieldStruct}}~{Rs}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3149fc73f3626eb90d8ffbc6259b266c543eaec7594c6c6c6503c7eab6bc76d6 \\
Pure Prop Address: & TMQXpz1uzda7WjR5D59goy6Npidv1Fh3ego \\
Theory Prop Id: & 638a457d1b7e3a0558d1e4a8349534f593bad615e24d92349d002ae3afb8f22e \\
Theory Prop Address: & TMGNWDRJRSAseCPGm2L1ufW5qeZCWTf1RTZ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:gejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgehpfbdgg} ${{({\uniddcgcg x y : {\iota} .  {{{{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{1}}~{x}}~{y}}})}={{\mname{RealsStruct\_plus}}~{Rs}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7affbde70101a9fb21231dde40bcdaf417f643cdb83ddc82db6176651037aac3 \\
Pure Prop Address: & TMe11GXyXEF6MGDABPzVWHtkhaqqa9Fkk9T \\
Theory Prop Id: & fe164da026cbaa682ef9fe1136650ba3b8eb25224ebc8d70d1da787a38e26a69 \\
Theory Prop Address: & TMPP93FXoNZbKN3CDWyZUbmssxCbWqaUC2x
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:gejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgehpfcdgg} ${{({\uniddcgcg x y : {\iota} .  {{{{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{2}}~{x}}~{y}}})}={{\mname{RealsStruct\_mult}}~{Rs}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f983dc416a105a88e0acb3737c63f1e0d73c3c64c4520277a6e8d01d56bc243c \\
Pure Prop Address: & TMGdwUUczg7R1h1T84CWJM1aykiAmFN7fNu \\
Theory Prop Id: & 5715158dd199c38a59bbf5c3c0b1f41806ecc17815efa3fc7d133801a3cddea5 \\
Theory Prop Address: & TMMmVJkECh7QnA46fbLjGoemcoeBWqcFETd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:fgihahmgjgdgjgehpfgejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgehpfcd} ${{{{{{\mname{explicit\_Field}}~{R}}~{({{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{3}})}}~{({{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{4}})}}~{({{\mname{decode\_b}}~{({{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{1}})}})}}~{({{\mname{decode\_b}}~{({{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{2}})}})}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & cedc05dfb417fc06394a10542007a2f8ce55cc82ac0946b195b2c73fdfe9ade2 \\
Pure Prop Address: & TMNT1KjdwrJ9LD8Vnb1Xouv3qSy7pD9ivJ4 \\
Theory Prop Id: & edd0df751b28b776ab98f509eb6e9906b9ee240bf87903c0e4f8d10355c8de2c \\
Theory Prop Address: & TMYv4TivoSVD8m2uZbe9Vwkjd9899mo2iPo
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:gejgfgmgegpfgejgfgmgegpfpgggpfcffgbgmgdhdfehchfhdgeh} ${{\mname{Field}}~{({{\mname{Field\_of\_RealsStruct}}~{Rs}})}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0ec36b9dac36f4bf382d817f90965f789c23b007329c1f2a14b0014a91d92067 \\
Pure Prop Address: & TMZyZoKXReKnGyDuwVdpxzHCHi3LLUXkp3d \\
Theory Prop Id: & 23c5218dab13d451320a4c008e8443b47261ff382e01d4a9eafa9b5032fd81d1 \\
Theory Prop Address: & TMR5Ubk1J3EVtZScLio9b8k4kf9rtQ6ncTX
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfkhfgchpgpfjeog} ${{zero} \in {R}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 005b3674102dca79680f3c66d82d0a566ec5ce1fbc87acc22a9a6ca9ebebf54b \\
Pure Prop Address: & TMMBmyfDd54QzeVfnKcj1zfp6a52pbqPYQi \\
Theory Prop Id: & 7813991fa3eec14bade682c678cb4927bd388c9ac91fba2e332415aa6f370b7f \\
Theory Prop Address: & TMcHAxvwy4L9sxYGy2zcQMnxHoHXr9htE8A
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpgogfgpfjeog} ${{one} \in {R}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1b33553dbfb8dd1c45ebfbfd7313edc634050f17c1b4547ca0b73c8154450951 \\
Pure Prop Address: & TMVFbWxvhhJfevVpKgMa2zFT5nstpC3SnzU \\
Theory Prop Id: & 04a7a006f820df56470471035fcc3ca727dad5b96142181cf37a3fa7867815f3 \\
Theory Prop Address: & TMQ7UNpiYfxP56HgBYDN27D2wfmUQoY6GXz
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfahmgfhdhpfdgmgpgdh} ${\unicdcdadad x y \in {R} .  {{{x}+{y}} \in {R}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d59fc8a19b30608e6133175e65740cbf406831dcc29d41398580efd3565b0304 \\
Pure Prop Address: & TMH8yov9d8x8DhvNqVPDNBwvVUPZbFnQY4N \\
Theory Prop Id: & d047b1443a58c7267b47a6534919e45b97226c58a703aad09b0fcb5a5e1649e2 \\
Theory Prop Address: & TMMHW3MKm5vVqaUcstZAAHdswmLybKpv7jC
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngfhmgehpfdgmgpgdh} ${\unicdcdadad x y \in {R} .  {{{x}*{y}} \in {R}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b521d3f1983a1d40a297548001ef03c96963cfbb20d3944063f920b35cb4f799 \\
Pure Prop Address: & TMZECePNUD8jFwZt6i2o5f1wf4rK94xSn89 \\
Theory Prop Id: & cd4e02b42f650e37ee913b313e6820d2ad0116d56cfd6e5ef595d72f2bc261d6 \\
Theory Prop Address: & TMbyKuyhM9cra9Gm7KfgFoAyUN1YdzMLr52
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfahmgfhdhpfbgdhdhpgdg} ${\unicdcdadad x y z \in {R} .  {{{x}+{({{y}+{z}})}}={{({{x}+{y}})}+{z}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f3b3129567a8db11a6326406f56b94419e2d9818eac0f622df322d09400a6bcb \\
Pure Prop Address: & TMJzLB6CoD1vut7yT35PBx1oo2PXUVc7Zig \\
Theory Prop Id: & 8f95762ccd9f93efb2b813048852b496d88e5771e670e7c8330941bc2870621b \\
Theory Prop Address: & TMTfndBh736RkfuCoAVb5EfY859TyC26KS3
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfahmgfhdhpfdgpgng} ${\unicdcdadad x y \in {R} .  {{{x}+{y}}={{y}+{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 59c94c4de8b2476d2510dd4b09ddc432083d2bf7b7fb33b2024269e7cfc66637 \\
Pure Prop Address: & TMduLqT57Q1QdbQNsUt6QSDeqMnyE1sLKgk \\
Theory Prop Id: & fe9682e564377a859f1f577491e17a474459086d1c02b7e016b4b3d5d708413c \\
Theory Prop Address: & TMdBQiCvauMBPkmwE1yJizNxoC35UqZBmtT
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfkhfgchpgpfme} ${\unicdcdadad x \in {R} .  {{{zero}+{x}}={x}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 941604f00ed1e86b155b1b1cc752ab549a904f9ec545ac4e5e0501edea6bf9ae \\
Pure Prop Address: & TMVWFuU6oaPoeDy7jmijYCvqgjMhDBfnAp6 \\
Theory Prop Id: & a028d8ecce4c4d08176730f488a7aa5d9e462795f24bc924a2776fee82844e1e \\
Theory Prop Address: & TMTKfSnfXd5nWfTXC5qtkJxtoSvW9Z1vQDx
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngfhmgehpfbgdhdhpgdg} ${\unicdcdadad x y z \in {R} .  {{{x}*{({{y}*{z}})}}={{({{x}*{y}})}*{z}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a3751ec23c16fcb964095d434bd62fbff3f54ea6f367daa73dd6af90e7e15285 \\
Pure Prop Address: & TMVx6KqoHebZKqDiuAYv9dmMFLDaUM7Cqcs \\
Theory Prop Id: & 9964f7eae81893737f8b8d14d0e37c2cbbffc7ba0883b2c17e9015b758de91f5 \\
Theory Prop Address: & TMHd9hTsZZPRjz6qz2H9V4kWSaeReQd6WyF
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngfhmgehpfdgpgng} ${\unicdcdadad x y \in {R} .  {{{x}*{y}}={{y}*{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 917b61b0f80ba1c680830ac638f010b6d7ab9948b61b6bf2fa194f1559175899 \\
Pure Prop Address: & TMYHM6VL7di1iR2UetLF2MsoT7BqwWRXefM \\
Theory Prop Id: & 56ee11d2a162693703d6398a6fa3b44e437c11cbc2034fd1d87a3616b79c425f \\
Theory Prop Address: & TMM9uq46vBF85FpZjdJoCTa8ouco6ezSTf2
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpgogfgpfogfgbhpfkhfgchpg} ${{one}\unicdcdgdad{zero}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a175b20ec46a263f0237a1f588ae8b080a36967feae498f91cceed32f1f07bfa \\
Pure Prop Address: & TMNKMBM19aBBHJYg384yRYAbPN53WVbc5ns \\
Theory Prop Id: & 52ab44a448083c215a50ab8191899ef26e1fc7da0b367bd95ecbe48ce8e0c5ac \\
Theory Prop Address: & TMPjniRmWmW6zxw8tiEspuizDHoc7XRo3VJ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpgogfgpfme} ${\unicdcdadad x \in {R} .  {{{one}*{x}}={x}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 25b4a10707d2aab2f4bcc2756fdd2e3abf00c1f705db1132de37a2fc6ea6f637 \\
Pure Prop Address: & TMMvdyQpdr8L3T42os9auY5Nm8FUhCGnw76 \\
Theory Prop Id: & a03ccb9f27af94b22542ec92ba2b8ba628d78ad3d57905ee68cf0d731ae618ef \\
Theory Prop Address: & TMXFnDHcsSjoFcQzEX5A9GawZJ2M6iZkEV4
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfegjgdhehchpfme} ${\unicdcdadad x y z \in {R} .  {{{x}*{({{y}+{z}})}}={{{x}*{y}}+{{x}*{z}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7f277f99738c86e4cd0c17ce27f55256d4b83bac8c3ec324b94d179eb523992c \\
Pure Prop Address: & TMMJoedUhPVrXxiACByyi6RA7wzPg2bP8AE \\
Theory Prop Id: & 80b5a5b1ce589268084c8d0ff372432a6afa48599f754381e6ecf72024d11b73 \\
Theory Prop Address: & TMUbkahieeSRSSdoPifi8CERkncX96vdrCA
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgfgbhpfchfgggmg} ${\unicdcdadad x \in {R} .  {{x}\unicdcdgded{x}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 868cecbda36c1ae46534769ebbdb84ef47482569c02f1c634ef3b569a7a12c94 \\
Pure Prop Address: & TMUGH3okNJP2QDn6HWyznB2itjVp93X4N2M \\
Theory Prop Id: & 16a3fb53c4f1ddec154a070ed94ab00c6d6f96a8618fb21ee9e7cf9b7c48e5b1 \\
Theory Prop Address: & TMNhQCrwQkgbWcYiN9bZNZmiH341kRj81fx
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgfgbhpfehchbg} ${\unicdcdadad x y z \in {R} .  {{{x}\unicdcdgded{y}}\unicdbdjdcd{{{y}\unicdcdgded{z}}\unicdbdjdcd{{x}\unicdcdgded{z}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 09ea87b2c8bb2a8ef1d5e4e3413f6ed17268c09df43a45348a5029aef49aec25 \\
Pure Prop Address: & TMWjFSH117sB2fX5RBJDyEzKLmyYTS2S17B \\
Theory Prop Id: & 31d22cb5ee22963edf60c65a1536469fb0185ae68587cf700c23504bb704d4a9 \\
Theory Prop Address: & TMUUyEqPQ9BoJF9RsDg9cXEhU8SBZvmB7dv
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgfgbhpfbgogehjgdhjhng} ${\unicdcdadad x y \in {R} .  {{{x}\unicdcdgded{y}}\unicdbdjdcd{{{y}\unicdcdgded{x}}\unicdbdjdcd{{x}={y}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 13d82edccef1b70d008353ac64c4821740a0880f70e0e680d356e6d9d079c525 \\
Pure Prop Address: & TMHzbKmyZCrxTuS3pfvy16rZJqPRa2zbWR4 \\
Theory Prop Id: & 0aa45faf2f614bd2d02eaabb71f298ba2905b9e906d084f527121abf9d477ea5 \\
Theory Prop Address: & TMNKwtGqpbPqDk86iVJJmWZZzeTFCETXo97
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgfgbhpfmgjgogfgbgch} ${\unicdcdadad x y \in {R} .  {{{x}\unicdcdgded{y}}\unicdcdcdid{{y}\unicdcdgded{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8492e0db1af1c27447e1da622672c190501a41929ea1a5d824d99bf5c02e2cd4 \\
Pure Prop Address: & TMJpogMX6MJSnbVEpoyobdqAxQAvxSa6593 \\
Theory Prop Id: & 1f84f249a4223c91c6d7a2a24ab7e57bb4917d1c8b23d477391ddfe94e41e9a0 \\
Theory Prop Address: & TMN3dF6E7pq8oupzx7bGMNiMPz5ir8mq112
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgfgbhpfahmgfhdh} ${\unicdcdadad x y z \in {R} .  {{{x}\unicdcdgded{y}}\unicdbdjdcd{{{x}+{z}}\unicdcdgded{{y}+{z}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8a4ab12e1315b912d8d64a150d7f14264405d48f8b09b19e281e9f7f5babc606 \\
Pure Prop Address: & TMcTUgJ6Y2J84VUxaEX5jmfrXQaXknJHnsS \\
Theory Prop Id: & 37b323e19944cb2094fbf59f40ee6dc75941a5e8638cd782e29f1595ba2feb8d \\
Theory Prop Address: & TMVdpCNvU6P7MxmUMqGyRQUPygZTUwsQj7w
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgehpfmgfgbh} ${\unicdcdadad x y \in {R} .  {{{x}<{y}}\unicdbdjdcd{{x}\unicdcdgded{y}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 034e19d8815d7cac710f94e707f400dcc18e80e521efb31f4f07c41cc7eb551e \\
Pure Prop Address: & TMRG27Re6ZfnLLrivhufYbciXkVrRqvBff7 \\
Theory Prop Id: & 8b5db0b565ec35e47e13bb761e5bf3d54bc087a8eea8a3da96873a105ea5b67d \\
Theory Prop Address: & TMRAfQdUMEz2W8xXfP2Fp3jgBADZyt3d5q9
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgehpfjgchchfggg} ${\unicdcdadad x \in {R} .  {\preoh{{({{x}<{x}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 718b2a7630e43c2aa1a7ba78859688706a27263cc0e14ac10b19d2502d90d0be \\
Pure Prop Address: & TMT3hWZ2i8mzcX5AV3nDJbM74z3kxuqnWTT \\
Theory Prop Id: & 2c2cd678622bfb74b4ccf1f72f60b44aaeda5e1673f8577488821e96d3567025 \\
Theory Prop Address: & TMGLCxFW2fPYxLxs6VnBV5otswJNkwPZmso
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgehpfmgfgbhpfbgdhjhng} ${\unicdcdadad x y \in {R} .  {{{x}<{y}}\unicdbdjdcd{\preoh{{({{y}\unicdcdgded{x}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 036716b7031d9e9545f672350877cdd3cd606d761131cc3ce1039205806864f1 \\
Pure Prop Address: & TMJhZqMGmM2ycUqQT2qouBhNpEHfUJNVWVH \\
Theory Prop Id: & 5cf7bdfe85d03b1346539e85cb1f326864fbad71343355b85a957631353d7744 \\
Theory Prop Address: & TMVm3xj3nwvohAUcnSzL6EYJStDpJiCLKGX
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgfgbhpfmgehpfbgdhjhng} ${\unicdcdadad x y \in {R} .  {{{x}\unicdcdgded{y}}\unicdbdjdcd{\preoh{{({{y}<{x}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2cf72677c60b93322ea19a0826dab959dc3147d59f5d9d3fe5aa8b7174ee2ba1 \\
Pure Prop Address: & TMPymWMasfNM2Rji1AUDQcQM4LrH9uWx3Va \\
Theory Prop Id: & eee248a189252e629bc193acfb88c7bba2acdb08489e25f8f10c512171761595 \\
Theory Prop Address: & TMcmYMME185KoSKgmuzQKpMQLXMeqigGBdb
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgehpfbgdhjhng} ${\unicdcdadad x y \in {R} .  {{{x}<{y}}\unicdbdjdcd{\preoh{{({{y}<{x}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d64488d8f76ef862d95e7112eece4f8a3f3c67e4b05abe79c8e71640fdc45d88 \\
Pure Prop Address: & TMViyVGCbgDmrEXs1auawYkVTQNQjDYXtNt \\
Theory Prop Id: & 95a9252b4e98f9f989fdb3147e32cecbad9ba3739992d1cda07bac3cf57c3017 \\
Theory Prop Address: & TMSE7BJFjaB53kpm9JzcemAjmx17n4miSVo
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgehpfmgfgbhpfehchbg} ${\unicdcdadad x y z \in {R} .  {{{x}<{y}}\unicdbdjdcd{{{y}\unicdcdgded{z}}\unicdbdjdcd{{x}<{z}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 27e379d62bccda6def02d503ec80dce5751e1b6c06ea0598c0e4f12313a0c0a4 \\
Pure Prop Address: & TMXQk3yFeunwa2tgkwDRkTAiC72gnc1kEbQ \\
Theory Prop Id: & 0ad1d49b39850f537c0804836720afa6e45354a2506c9efdf082c5b1daef964a \\
Theory Prop Address: & TMYGBds2bCxGu39zPdjB8uTHjqLijWHi2KU
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgfgbhpfmgehpfehchbg} ${\unicdcdadad x y z \in {R} .  {{{x}\unicdcdgded{y}}\unicdbdjdcd{{{y}<{z}}\unicdbdjdcd{{x}<{z}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a7fac80d5d695102c89d549a0ac57e21a1da6b01abfff6821d8671d05ac5bfe9 \\
Pure Prop Address: & TMWRFzWY9Ks96UCp5hHDSSwUHcp2kHabMiZ \\
Theory Prop Id: & 17dbb16615d06d0e91aeb16bd8d187a094d35156dfc10c9a0cb26df2949b1a10 \\
Theory Prop Address: & TMTNZLqjCyTkXHcJDFnDct7KWxJxFjE1gTZ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgehpfehchbg} ${\unicdcdadad x y z \in {R} .  {{{x}<{y}}\unicdbdjdcd{{{y}<{z}}\unicdbdjdcd{{x}<{z}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & db51e413c8dae5e58659c817cad674fc9603959ac8908d0d3173b2f974bf800b \\
Pure Prop Address: & TMbyGzMwnbYf7Ef87fxfadSAny6Pk5qsDtt \\
Theory Prop Id: & a20edd4147dc9434a2c2a9b874d21178efbdb0a783f8456ad92be39a4e9cc999 \\
Theory Prop Address: & TMcywXpnEowovbp6p7WFLjPSpRAsCXmnz7r
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgehpfehchjgdgigpfjgngahchfgeg} ${\unicdcdadad x y \in {R} .  {\unicdcdadad p : {o} .  {{({{{x}<{y}}\unicdbdjdcd{p}})}\unicdbdjdcd{{({{{x}={y}}\unicdbdjdcd{p}})}\unicdbdjdcd{{({{{y}<{x}}\unicdbdjdcd{p}})}\unicdbdjdcd{p}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 94de406929eca9f0fbfb4a2208de46b59aa03b0f73d1c028a0551f7cff85ac9c \\
Pure Prop Address: & TMPYj9ARJvgSpTSMs4ztnxZZeSPKhuoUoz5 \\
Theory Prop Id: & f9cff737117c67a0fa5ded42ee2d5363ce2958c66b29c20fc47c132e3eb9efdd \\
Theory Prop Address: & TMF5ZzdDFzFLMP7TYe4pY4t96duPiB7Gv7f
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgehpfehchjgdgig} ${\unicdcdadad x y \in {R} .  {{{{x}<{y}}\unicdcdcdid{{x}={y}}}\unicdcdcdid{{y}<{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9b61418aa95cad0821b7b70b1ea6c48b07b043fc6317f9c66ece033f5aa59bf6 \\
Pure Prop Address: & TMboWZEUUV8p5yiVECUimfWvDD1YMWBhHDh \\
Theory Prop Id: & 65d4e63207e42cdaab1884c4d8cdfd112007705d830e2a0ccb44f59f328c8223 \\
Theory Prop Address: & TMbrq68wUob61FbNk4jqcD83WaJgVRjAJoe
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgfgbhpfmgehpfmgjgogfgbgch} ${\unicdcdadad x y \in {R} .  {{{x}\unicdcdgded{y}}\unicdcdcdid{{y}<{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b5d0e7fc3cc6bce2c2a17bc42c6f157d981ae1ed7c08efd6ac7ca23f0356d3f0 \\
Pure Prop Address: & TMXnJJbrRwtujq4xzLDYXsAr5wYR1aqSr18 \\
Theory Prop Id: & 7c7a8089b86844735c6b2d3f4a519d71e25477cdc5a960264b601472a84efadc \\
Theory Prop Address: & TMEmCCr1R4uUv68FnD6w9CyYspAKC65Udc3
\end{tabular}
\end{center}

\end{theorem}
{\bf{Notation.}} We use $\prenc{-}$ as a prefix operator  corresponding to applying term ${{\mname{Field\_minus}}~{({{\mname{Field\_of\_RealsStruct}}~{Rs}})}}$.
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpffgbh} ${\unicdcdadad x \in {R} .  {{\prenc{{x}}}={{{{{{{\mname{explicit\_Field\_minus}}~{R}}~{({{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{3}})}}~{({{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{4}})}}~{({{\mname{decode\_b}}~{({{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{1}})}})}}~{({{\mname{decode\_b}}~{({{{\mname{Field\_of\_RealsStruct}}~{Rs}}~{2}})}})}}~{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 47bc6dfa278b5babc8da67e3a1e0b527ec1d23809b12328023a26ca5c7249583 \\
Pure Prop Address: & TMYZAfb32ApJmMaB3PnEm7Vj4oTBY46NBAk \\
Theory Prop Id: & 66a3bbfbbb3361050ff6013b6efced8992a5edba8cbd9899558ae7cc61dbb8a6 \\
Theory Prop Address: & TMS87Bxs4mXuUt9jfG6JifxZNzvF7zzmfJt
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfdgmgpgdh} ${\unicdcdadad x \in {R} .  {{\prenc{{x}}} \in {R}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fdd8e0f6e300511edced7df68d989a615e67aa1bc7de750f9a619a82debfc807 \\
Pure Prop Address: & TMWr5eXBuGtfE7oLQbomsUzc6fjwmfid258 \\
Theory Prop Id: & b7c738e349ab2d6f6e8ad276adafcf81dc9aca3628028a9f35cb26fc2d5a34ef \\
Theory Prop Address: & TMVd1PVi5iMBiKL7rtZswTGDLj8iQbjZCSh
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfcf} ${\unicdcdadad x \in {R} .  {{{x}+{\prenc{{x}}}}={zero}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 09095b96433691485a96edf8101177ad9b033cef032350fa0ce5069fae93de9f \\
Pure Prop Address: & TMLSWdagYu788EzMjhuZP6Qm21mcP7gao1q \\
Theory Prop Id: & d7fd806b6ae1b4b46446aada0c9c8d453b985568eb38cc1ba86372ba219d557d \\
Theory Prop Address: & TMMMVeZuEoKYrJsCZYGX7isAfmKNaheKNC6
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfme} ${\unicdcdadad x \in {R} .  {{{\prenc{{x}}}+{x}}={zero}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 98ac5d2df769fab37af56d31b7a7b0a24ac969c2c72d9889077705d5c24ceabf \\
Pure Prop Address: & TMKzv2RXJzXuZKiMMKDdny2ePWZafZ7U5ZX \\
Theory Prop Id: & 23218f089518e8a515bb2026967940f961d8643db6282c71fb89bbac1b93ce39 \\
Theory Prop Address: & TMLhepvAwS8gbHygkWi1J1B7BRAUEkXdFje
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfahmgfhdhpfdgbgogdgfgmgme} ${\unicdcdadad x y z \in {R} .  {{{{x}+{y}}={{x}+{z}}}\unicdbdjdcd{{y}={z}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5ff5748a79dfce201548e0ea8e7fc3e8ed66fae54c0079d8409b468ec617420e \\
Pure Prop Address: & TMXHN9o2PJFBMaNmp83BfCNYAonHHWiGFzX \\
Theory Prop Id: & ff6953bf63e902e9883d98d6d01c20077d6e823216570f791adbe886fb641882 \\
Theory Prop Address: & TMcrA1J9U6fRWwNyvbHM8Pvu7xQqVhBf1Vp
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpffgbhcd} ${\unicdcdadad x \in {R} .  {{\prenc{{x}}}={{{{{{{\mname{explicit\_Field\_minus}}~{R}}~{zero}}~{one}}~{({{\mname{RealsStruct\_plus}}~{Rs}})}}~{({{\mname{RealsStruct\_mult}}~{Rs}})}}~{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e8ae10f25c6af12d1958559a4f188545a0cb40752c041c3f8d11eecc9bcd7199 \\
Pure Prop Address: & TMTjWE5eDPyLEbX1dfg7EizwEtFGXzUQkDt \\
Theory Prop Id: & a1211f03836b5eee939c778de3c1018416ca8ed1d3ee2b73d2266429b7c3e3e2 \\
Theory Prop Address: & TMaFALex8WPZq7Zo2cyTerKK2nzDvTo1quC
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfahmgfhdhpfdgbgogdgfgmgcf} ${\unicdcdadad x y z \in {R} .  {{{{x}+{z}}={{y}+{z}}}\unicdbdjdcd{{x}={y}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & baf30bc9e3bd74b52dd88cb1eea019d239e8ac999133603bb6c0ffe240f0585d \\
Pure Prop Address: & TMJ8DeDPrVjLKi4AWwmCV15PJt4zh9Qea36 \\
Theory Prop Id: & a5736933cee0a9aaae630453df6f585065561f78e9b7df048786845e7dddfdc1 \\
Theory Prop Address: & TMRycJqeTxMBEZCvL8J2Hpzh8k9oRbHARWu
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfjgogghpgmg} ${\unicdcdadad x \in {R} .  {{\prenc{{\prenc{{x}}}}}={x}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 08c894cf89246f6d453037ab7581b8a4980b918261d561476b8d1c569c9972f4 \\
Pure Prop Address: & TMZQm7b2b6rZPGFcUA5n5jUA5xm2TjgdNtJ \\
Theory Prop Id: & 58988336a2e2c10ce0f8c945854017783f1fdb55c1a40daf0cdb8f1a17da8153 \\
Theory Prop Address: & TMExht3DFRB3GYdQvpgYztYDMSMkBYJGNpx
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfpgogfgpfjeog} ${{\prenc{{one}}} \in {R}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & bd1d4f7909be8f917eab926743f5502922a8a810a93cbd00d6412952a908801c \\
Pure Prop Address: & TMWc1PSLG2857QLtbTuiWKgfqKfe8q9nLcz \\
Theory Prop Id: & 39bedc18953a944c01c33894b19ee20e55993b517d311d2daa46cd1165c05d3b \\
Theory Prop Address: & TMR42g5o3sV9VSPe54YRCdkizri2QPBCjT6
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfkhfgchpgpfngfhmgehcf} ${\unicdcdadad x \in {R} .  {{{x}*{zero}}={zero}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e8baa64a646894666f2b0b82b5918f7ed9318f8f8eee4f59dbbfba3c20d9ff21 \\
Pure Prop Address: & TMPdW3hMJTztdGUdkDRakYTSjbabprAc9Z6 \\
Theory Prop Id: & 88201a486135db74001498754cf27b8685565cca99912ee888c9482032e13640 \\
Theory Prop Address: & TMX7vtUzERwfALngARjEZ8VQG9g4HjUN3xS
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfkhfgchpgpfngfhmgehme} ${\unicdcdadad x \in {R} .  {{{zero}*{x}}={zero}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 751db0caf92b3fe28e704ad517da9de59bb6d3b21c4e04d1f9b4a77d601f5c33 \\
Pure Prop Address: & TMH5oywcudUDCPXjnkAtprfxnT8Mx3M4znP \\
Theory Prop Id: & 15727a520c07b04ff12d5865f874e26ce2c8daf0a2613a88bd45313469d1bb82 \\
Theory Prop Address: & TMU6ughQfPFJrRB8wkmoxv7fyit1ENwF82x
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfngfhmgeh} ${\unicdcdadad x \in {R} .  {{\prenc{{x}}}={{({\prenc{{one}}})}*{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d488fae83059b5b9d5f6500d2855e253c82b63de6ffcbcc7a9a08a9e90364a54 \\
Pure Prop Address: & TMYWs7B7tsm6BLxQZXktxsTeJ7uiY9hTMj7 \\
Theory Prop Id: & 221fb8b61d0263c90b7a1b788de3aeb55c2d1c2061e15c7cee6e1abf97657d4b \\
Theory Prop Address: & TMXHd2hW4LazoJxHtfBqjWYQHGqzvfC1yW7
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfpgogfgpfdhbhfhbgchfg} ${{{({\prenc{{one}}})}*{({\prenc{{one}}})}}={one}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & cab4aa58bf0267bee8cf3c2536213ef66d714059ef2218d7e6f1728f28f8ee99 \\
Pure Prop Address: & TMPnEwkVaRzmRUkEJ1woHFG1DZk5uMWQaEU \\
Theory Prop Id: & 701edc74dbc3b46a566cc9743166e0ab300f6a34691eebe4339c03df61b41a75 \\
Theory Prop Address: & TMQvuBviqrrzJPYbWJNSTnSuiYbaqYvrUro
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfdhbhfhbgchfg} ${\unicdcdadad x \in {R} .  {{{({\prenc{{x}}})}*{({\prenc{{x}}})}}={{x}*{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 95c7432ee6391c9809080d41eed33f798d0d274f825cb91bde1c81c29c2d0fb2 \\
Pure Prop Address: & TMb7xqhMnUxosCkgLM9VrW3WNqF4se2wayn \\
Theory Prop Id: & 93bb8340ea11964abd12d112440b8853a2dbc1264d1238cd0571b528ea47232d \\
Theory Prop Address: & TMaYV6vuKLDWHanfeHhxV3h8HcDTY2T7wrM
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfkhfgchpg} ${{\prenc{{zero}}}={zero}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a65ac7f737bccdfc3753590b8bce334772b3b6919a1922509b3d0063820c713e \\
Pure Prop Address: & TMTqKgfPgRZixBhk5JsKaM8ZeTqva2dbwrH \\
Theory Prop Id: & 31d61336fe6999362705f5feadf4f373b68c3c731a34d56603a78e289700a6fd \\
Theory Prop Address: & TMYiSLxX166trdrKLyA51oNquqKN3M3YbqS
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfegjgdhehpfcf} ${\unicdcdadad x y z \in {R} .  {{{({{x}+{y}})}*{z}}={{{x}*{z}}+{{y}*{z}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 862fd843939bd3632046f172b4ceaf3d4e6b2698b28cb880f14e41743a84cf8f \\
Pure Prop Address: & TMGHAMzokV28j6GZ5fDvRnCYLM1h8jYg2jh \\
Theory Prop Id: & ce3537a1b799f75f41b619dd337ad2faa0ab92ffa22088d81b931f86215f45cb \\
Theory Prop Address: & TMc4SuM9sH5ftzoRUZc5jyapjLYJmbwrTMH
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfahmgfhdhpfegjgdheh} ${\unicdcdadad x y \in {R} .  {{\prenc{{({{x}+{y}})}}}={{\prenc{{x}}}+{\prenc{{y}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & bbbec91faa7e9e78adb2960de270de19dc98814dd4909c4e5f8279063acf2549 \\
Pure Prop Address: & TMEpdMaKhWF6yhGZvGSnnsfdxW3JE2HhaX7 \\
Theory Prop Id: & e102afbd1d47d23a82a7d28fb39b18fed4f9d1368508a7d3169ed416e1e3ecac \\
Theory Prop Address: & TMXSKFTjeXzq6D3jVPWKqcXCcbSQehf31Mv
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfngfhmgehpfme} ${\unicdcdadad x y \in {R} .  {{{({\prenc{{x}}})}*{y}}={\prenc{{({{x}*{y}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 87a9ee20b9e8f4696f85fe723b6595c3796cd59ace6d3cc05a2f8d4d42e69893 \\
Pure Prop Address: & TMccDiAHv34RsKwiFv5Ze6HutuaTy3ivq71 \\
Theory Prop Id: & 145734116522a560429224bd70db12b8ab512947aa0e49a218ed22e31a0e7111 \\
Theory Prop Address: & TMTFyy4RHjnYmm6dmcA1XLwxqwZju5r6roQ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfngfhmgehpfcf} ${\unicdcdadad x y \in {R} .  {{{x}*{({\prenc{{y}}})}}={\prenc{{({{x}*{y}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1dfc6deb04af278e1661d3df5a3bdea0daab232c9302727cd688b9c76e430dbc \\
Pure Prop Address: & TMXtdQduGF54FwdCwf1pD3PBBy4rK8RihuY \\
Theory Prop Id: & 8cd9e0d875d43b1ddc9c929156b184121abe145cc15e0d102566046cd722f439 \\
Theory Prop Address: & TMSbRFVgjqs3EKV1FxZLbyQFWKwbK5ZxJ6C
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngfhmgehpfkhfgchpgpfjgoggh} ${\unicdcdadad x y \in {R} .  {{{{x}*{y}}={zero}}\unicdbdjdcd{{{x}={zero}}\unicdcdcdid{{y}={zero}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 52e521e4ea57136109ff9c68732a22d7e0fce2c81fe7e0333f27725db442259e \\
Pure Prop Address: & TMSLZXtw9SNRfYRMKsJi6bSZm5LUozCpXP8 \\
Theory Prop Id: & ca0065eef41293f5e9cd0d9a5017dcd5aa4f3456b0df6f2e56a71ae1b77b12ff \\
Theory Prop Address: & TMT3MLk7YqUD9iX8atswcLFDca5cKNVBF5B
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfdhbhfhbgchfgpfkhfgchpgpfjgoggh} ${\unicdcdadad x \in {R} .  {{{{x}*{x}}={zero}}\unicdbdjdcd{{x}={zero}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8aa0cd6d0c435ca74fff47f90162b7c2c3d6b793b431dc4b57459872f365c838 \\
Pure Prop Address: & TMaEmRhRXsGfs29zHkD7Vcbp4X5kNcSAapY \\
Theory Prop Id: & d4c42484e082d273bd7d0b5e37d7d640a925b2221ce532e2e8bf25c52fb9e418 \\
Theory Prop Address: & TMNUaVt1RUxY5s8TcGo5CqbLi9uuLiivewh
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngjgogfhdhpfmgfgbh} ${\unicdcdadad x y \in {R} .  {{{x}\unicdcdgded{y}}\unicdbdjdcd{{\prenc{{y}}}\unicdcdgded{\prenc{{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f49a3223293a899b90e3f0e2b2ccb4c7c143e078381e7b41b18f20349f7da516 \\
Pure Prop Address: & TMGUQNAZwd9CygmSnCMdCF9RyBCaAERrmJ3 \\
Theory Prop Id: & 56adcd399875c337ec8eb70665c36638054ac2e7b9ce43b5d38d0fd6d09d41fa \\
Theory Prop Address: & TMMV3gE1eKPX49sr7atFfzx8pAxwxmwav9U
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfdhbhfhbgchfgpfogpgogogfghg} ${\unicdcdadad x \in {R} .  {{zero}\unicdcdgded{{x}*{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1f7e7600170f3d0edf48eb23103a7e83310e8bcbbb489afe89d2a550424bc433 \\
Pure Prop Address: & TMX92WctujuKb5AX5z7JnBAbwpRUF1F1Yrn \\
Theory Prop Id: & 116beb08289e1dae25c8f46eb411554c9409ef2e3486a7b37ca72edeeaa82c18 \\
Theory Prop Address: & TMHDYtHgvcDVHrwczd4cftPoYYYa2bG9w9x
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfdhfhngpfdhbhfhbgchfgdhpfogpgogogfghg} ${\unicdcdadad x y \in {R} .  {{zero}\unicdcdgded{{{x}*{x}}+{{y}*{y}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 55dc2f3f44253c88b10d4b601b85eff9350d5214097a99dfdfbb9ed33ba8e291 \\
Pure Prop Address: & TMFA9PQVdmjfUj74vevkFTa7eHQkM1Vw17x \\
Theory Prop Id: & e739ff92218b151c98245ecc8ad2ef578e165ca921eee5b84841fa4b11c3c743 \\
Theory Prop Address: & TMSDysdaJ68HZgUqpWWzxnY91V1BU9Eprtp
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfdhfhngpfogpgogogfghgpfkhfgchpgpfjgoggh} ${\unicdcdadad x y \in {R} .  {{{zero}\unicdcdgded{x}}\unicdbdjdcd{{{zero}\unicdcdgded{y}}\unicdbdjdcd{{{{x}+{y}}={zero}}\unicdbdjdcd{{{x}={zero}}\unicdcdcdhd{{y}={zero}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a1f474df8b36a92e0b58e7c0515850cfcf881b7ddb216e72d98b55d27f765f6b \\
Pure Prop Address: & TMQqUDVayaoT53ydRzz8cF1HJtqHjdQhbha \\
Theory Prop Id: & 789223bfb56d1d985a4558cb85e56b74044e5dfa7a47009d14691aa59d1c5e97 \\
Theory Prop Address: & TMSUNEckz8DH6CXDWbVK7G39vzxmKYfFe5u
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfdhfhngpfdhbhfhbgchfgdhpfkhfgchpgpfjgoggh} ${\unicdcdadad x y \in {R} .  {{{{{x}*{x}}+{{y}*{y}}}={zero}}\unicdbdjdcd{{{x}={zero}}\unicdcdcdhd{{y}={zero}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 64bfc4ce2dead3e98b809e881e851e8a7b8d4a5992eef4830fcbb2460cb33ae9 \\
Pure Prop Address: & TMb4XesQGWXCWm9yHHnypZ9bBes3f2tF3D7 \\
Theory Prop Id: & c2b9c84106f6aa374eab5069dd30d750baf9110b257653b6b4701e79424f840f \\
Theory Prop Address: & TMcj8zjWByWFUKfZtiUpAAS1zrBiXhL76KL
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfmgfgbhpfkhfgchpgpfpgogfg} ${{zero}\unicdcdgded{one}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 75a9e5ea0655f6a7b81f9562d536cc979c304fee58c31e40455766da8a0b8491 \\
Pure Prop Address: & TMdQ75HNiDrTsVaaDbMqwf25EtKrdnktWAd \\
Theory Prop Id: & fc79c3810891aeb6dd0f786541364fa69b5d2c6cc14b14c578108424308f783f \\
Theory Prop Address: & TMWFE2LGFJYTijtjpZrCpn7C7MW7CXxMUks
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfoe}
$\mname{RealsStruct\_N}$ is the opaque object of type ${\iota}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 5e5ba235d9b0b40c084850811a514efd2e7f76df45b4ca1be43ba33c41356d3b \\
Pure Object Address: & TMbcMfzuQfUfq11QLM6oT11DkR9A7LKHyxL \\
Theory Object Id: & 49868cd01ccaaa52966256a5255beef5cfc5d2b96374f80d5069a1170c2c4c78 \\
Theory Object Address: & TMKcBDt3UC9DqpmqhzUAWU2YJ3EuMHE5yEh
\end{tabular}
\end{center}

\end{definition}
Let $N$ be ${\mname{RealsStruct\_N}}$.
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfbechdgig} ${\unicdcdadad x y \in {R} .  {{{zero}<{x}}\unicdbdjdcd{{{zero}\unicdcdgded{y}}\unicdbdjdcd{\unicdcdaddd n \in {N} .  {{y}\unicdcdgded{{n}*{x}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3d2ba33455345a9c0ba6e7d86b15bcead0c25d5b93f9dfdb5a52b615ac9babfb \\
Pure Prop Address: & TMWCft8x2BTk4nokUerxJK2X11JdeiVv565 \\
Theory Prop Id: & 1f0d5fe981bc213e8f88cb7c74a9bf90e20b53b02288f897c78dfc099ab3dd61 \\
Theory Prop Address: & TMZGPa1mwgbVzkFdodAv6RjSmb9g3Bhh3hq
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfdepgngahmg} ${\unicdcdadad a b \in {{R}^{{N}}} .  {{({\unicdcdadad n \in {N} .  {{{{{a}~{n}}\unicdcdgded{{b}~{n}}}\unicdcdcdhd{{{a}~{n}}\unicdcdgded{{a}~{({{n}+{one}})}}}}\unicdcdcdhd{{{b}~{({{n}+{one}})}}\unicdcdgded{{b}~{n}}}}})}\unicdbdjdcd{\unicdcdaddd x \in {R} .  {\unicdcdadad n \in {N} .  {{{{a}~{n}}\unicdcdgded{x}}\unicdcdcdhd{{x}\unicdcdgded{{b}~{n}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6a20f5ecaafcc7e432b7c8ac16d2d71517ad446546abe4f692d40edd5f175aab \\
Pure Prop Address: & TMSCeZV9tj3Dc22WnMpqqUpBYCFvYAsqBkc \\
Theory Prop Id: & bf5231ea148f4c4a8e2c8b3b3d3ac74a7a34a6f05496134d190b71e9d903d2eb \\
Theory Prop Address: & TMZEYVRwMZfXMAwxS5gUKffKiFvE5u1BQFx
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfogbgehpeggpechegfgchfgeggejgfgmgeg} ${{{{\mname{explicit\_Nats}}~{N}}~{zero}}~{({\uniddcgcg m .  {{m}+{one}}})}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6b43192d1dea3e3fc85a6591e66c6a726082864fd516780b26a754b19d6118ef \\
Pure Prop Address: & TMK4NwKLMJh9YDNuNjVZk7onpb3hJRVZcqP \\
Theory Prop Id: & 49002473fe506d6b9c9c98fcd03504abba2716d63b6a552bc1c8b880dc2511ed \\
Theory Prop Address: & TMUHoaa1stUMPCnj1neeSaXtn79MZ9CQ2cT
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfoeahpgdh}
We define $\mname{RealsStruct\_Npos}$ to be ${\{n \in {N}|{{n}\unicdcdgdad{zero}}\}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 18dcd68b13ac8ef27a4d8dfec8909abfa78ffbb33539bea51d3397809f365642 \\
Pure Object Address: & TMXGAJt8GRxM5LRMTFpsj1piXUY7f9nuK7P \\
Theory Object Id: & f5d38003115cfc06f11ecf0699e5b6ecab522d53550fce92c2ecdc20371c12e5 \\
Theory Object Address: & TMMXxyZsUkoJFF2GqJHfHwyksqYKy5LeJWe
\end{tabular}
\end{center}

\end{definition}
Let $Npos$ be ${\mname{RealsStruct\_Npos}}$.
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfafpgdhoebgehdhpfogbgehpeggpechegfgchfgeggejgfgmgeg} ${{{{\mname{explicit\_Nats}}~{Npos}}~{one}}~{({\uniddcgcg m .  {{m}+{one}}})}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 608937f38d90199679a353df12d26e9c24924a2849bcb600a52f62801cf8c8b1 \\
Pure Prop Address: & TMH3THbHi69Lk6JSBHKSnCQykxJcsLKkScy \\
Theory Prop Id: & 7be7ef55dfa7ee734330d79a5f088736665cccd7e5615ce026a8457ca9690f34 \\
Theory Prop Address: & TMUwcaA4tPYCgfSCsdNGLV1J2Wj2UjJKYcT
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfkf}
$\mname{RealsStruct\_Z}$ is the opaque object of type ${\iota}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 736c836746de74397a8fa69b6bbd46fc21a6b3f1430f6e4ae87857bf6f077989 \\
Pure Object Address: & TMZnqja29W4HpFDK7otUSfGC3Eomh9qtUB8 \\
Theory Object Id: & 0610d20138c846183403e57b8540fd87c9759a321e2a636ed8b592691ebcc810 \\
Theory Object Address: & TMUL7MYE83ADM6VaAkyBjtEUt5rRp6pVBu3
\end{tabular}
\end{center}

\end{definition}
Let $Z$ be ${\mname{RealsStruct\_Z}}$.
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfbf}
$\mname{RealsStruct\_Q}$ is the opaque object of type ${\iota}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 255c25547da742d6085a5308f204f5b90d6ba5991863cf0b3e4036ef74ee35a2 \\
Pure Object Address: & TMQNHtJRR1iyQCK2yVPcXNmgjvYt4LJWjPH \\
Theory Object Id: & 0a4db083012656ded8a542881fa97da3998c2c25649b64f9c88525f48f727d53 \\
Theory Object Address: & TMHUMp4nRyqMcV6TWWc9HRr7HJygDDY4KSh
\end{tabular}
\end{center}

\end{definition}
Let $Q$ be ${\mname{RealsStruct\_Q}}$.
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfoeahpgdhpfahchpgahdh} ${\unicdcdadad p : {o} .  {{({{{Npos} \subseteq {R}}\unicdbdjdcd{{{{{\mname{explicit\_Nats}}~{Npos}}~{one}}~{({\uniddcgcg m .  {{m}+{one}}})}}\unicdbdjdcd{{{one} \in {Npos}}\unicdbdjdcd{{({\unicdcdadad m \in {Npos} .  {{{m}+{one}}\unicdcdgdad{one}}})}\unicdbdjdcd{{({\unicdcdadad m \in {Npos} .  {\unicdcdadad q : {{\iota}\unicdbdjdcd{o}} .  {{{q}~{one}}\unicdbdjdcd{{({\unicdcdadad n \in {Npos} .  {{q}~{({{n}+{one}})}}})}\unicdbdjdcd{{q}~{m}}}}}})}\unicdbdjdcd{{({\unicdcdadad n m \in {Npos} .  {{{{{{{\mname{explicit\_Nats\_one\_plus}}~{Npos}}~{one}}~{({\uniddcgcg m .  {{m}+{one}}})}}~{n}}~{m}}={{n}+{m}}}})}\unicdbdjdcd{{({\unicdcdadad n m \in {Npos} .  {{{{{{{\mname{explicit\_Nats\_one\_mult}}~{Npos}}~{one}}~{({\uniddcgcg m .  {{m}+{one}}})}}~{n}}~{m}}={{n}*{m}}}})}\unicdbdjdcd{{({\unicdcdadad n m \in {Npos} .  {{{n}+{m}} \in {Npos}}})}\unicdbdjdcd{{({\unicdcdadad n m \in {Npos} .  {{{n}*{m}} \in {Npos}}})}\unicdbdjdcd{p}}}}}}}}}})}\unicdbdjdcd{p}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ac462c732947010e996e2195f9457d22c91ea98709503c70dfd886bb0ff05a86 \\
Pure Prop Address: & TMUvjp5vzLmJS9vokz3WxDwgYtiFC9YfGot \\
Theory Prop Id: & a2de5a1838f8c2dedd49986ce8e9b16887f479df3ea69491847fdc1ff939d7e0 \\
Theory Prop Address: & TMLHJgX12h6ob8HcMEAHsRmu8Sv6z8hA4Vg
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfoeahpgdhpfcf} ${{Npos} \subseteq {R}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 008a7ec9f81b901d173d4bb67cd3b9f8fd59e4f1d6086a46e01df9b0379a24ac \\
Pure Prop Address: & TMJ4VGyP4MLVJHuj3qvzEBau5k5S6d87mJp \\
Theory Prop Id: & fdd42c49268c6b0f74f093dd11460a4c3f87341db2ac930dc62f4e370c959def \\
Theory Prop Address: & TMFUveydbCfNR11n2X6XYqhXH9GGB3eCa7E
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpgogfgpfoeahpgdh} ${{one} \in {Npos}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 060e811ee11a2fdd622c33063c392284ab0e32d1798a3c22f69b0e2cc87e0bb3 \\
Pure Prop Address: & TMVQ8Vzx4rfk9PgrXA5DC8bupwA7Rc71ciY \\
Theory Prop Id: & adbd044729f5641d3c3402b8dae736e674669ab85664a907a855a441004cb866 \\
Theory Prop Address: & TMV8pdo8evmeKAvbv1pzGuyB3FWK7iaDugd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfkfpfahchpgahdh} ${\unicdcdadad p : {o} .  {{({{({\unicdcdadad n \in {Npos} .  {{\prenc{{n}}} \in {Z}}})}\unicdbdjdcd{{{zero} \in {Z}}\unicdbdjdcd{{{Npos} \subseteq {Z}}\unicdbdjdcd{{{Z} \subseteq {R}}\unicdbdjdcd{{({\unicdcdadad n \in {Z} .  {\unicdcdadad q : {o} .  {{({{{\prenc{{n}}} \in {Npos}}\unicdbdjdcd{q}})}\unicdbdjdcd{{({{{n}={zero}}\unicdbdjdcd{q}})}\unicdbdjdcd{{({{{n} \in {Npos}}\unicdbdjdcd{q}})}\unicdbdjdcd{q}}}}}})}\unicdbdjdcd{{{one} \in {Z}}\unicdbdjdcd{{{\prenc{{one}}} \in {Z}}\unicdbdjdcd{{({\unicdcdadad m \in {Z} .  {{\prenc{{m}}} \in {Z}}})}\unicdbdjdcd{{({\unicdcdadad n m \in {Z} .  {{{n}+{m}} \in {Z}}})}\unicdbdjdcd{{({\unicdcdadad n m \in {Z} .  {{{n}*{m}} \in {Z}}})}\unicdbdjdcd{p}}}}}}}}}}})}\unicdbdjdcd{p}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 017fe94adff66febe9607cfa8e37ab6ccbb7b674ee4fb54a05fdde026adcf09d \\
Pure Prop Address: & TMSzNh6NafcQ7JF4kYewkBLEW5kuczQKhgd \\
Theory Prop Id: & 5f0a7e775f83d554f276621ef295027944bb5559b7afbb223b8ddfc6b44e546b \\
Theory Prop Address: & TMQPzwy4S512aDpt1mBbU3fhUfvi6C5QvWZ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfogfghgpfkf} ${\unicdcdadad n \in {Npos} .  {{\prenc{{n}}} \in {Z}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f9e62d63a307ece18bbd80aa48a23f6f4d17cc93e2d4ced458eb45a3505e2639 \\
Pure Prop Address: & TMZmeurK9YG3FXNye9gkxqjf5hSGk8queat \\
Theory Prop Id: & 4d43ebd17d60a85f9c79d798f7b371ae521f4a7bff8c51313295f609080b380a \\
Theory Prop Address: & TMbm6VFrtyUyFrUCb5N5iv3gJ2ebnXd96kk
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfkhfgchpgpfkf} ${{zero} \in {Z}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e63b5bd24a1e20be4e359c4763b33e6b62e4af0ee2eb346b98a874d8b75cdf72 \\
Pure Prop Address: & TMZZPadL877aUv7uy4BJ4ZVoUgXHzpSDS2p \\
Theory Prop Id: & d617fb71b2447bb5050839c73d379af276584315cac62d0fc944a5d871c873b7 \\
Theory Prop Address: & TMQkD9oz6MNGEZp6SBtWuDTTfrftxMyra1p
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfoeahpgdhpfkf} ${{Npos} \subseteq {Z}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3b82b4daba391f17879dad1f437cec7ff368e07aaae1e064ae35d652ed72ec7b \\
Pure Prop Address: & TMHtwXjtrWbWqU2d751pwq11Lj48hsv1V9i \\
Theory Prop Id: & 654198ba4994329078fbb885ead1adc3e09120276bfd4095fccaf00cd2c50a5f \\
Theory Prop Address: & TMQdysxG2P8Lkp3CTWLhBkp2LqNBQWWr6Ak
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfkfpfcf} ${{Z} \subseteq {R}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & afb222581fed3232153f5f2d44da3bbf0ce181c57357472cf15a82a3d810681e \\
Pure Prop Address: & TMPwERt3rdFJJxsDnxdhcQgF2MpMFsf3i1V \\
Theory Prop Id: & 9b0ab1796c671927fbee291f23eb65871f651953d2f84aa13bc10861a90fcac1 \\
Theory Prop Address: & TMGkAuVyoZcR48zuQfsrquFkDgtyiNcdENZ
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfbfpfahchpgahdh} ${\unicdcdadad p : {o} .  {{({{{Q} \subseteq {R}}\unicdbdjdcd{{({\unicdcdadad x \in {Q} .  {\unicdcdadad q : {o} .  {{({{{x} \in {R}}\unicdbdjdcd{\unicdcdadad n \in {Z} .  {\unicdcdadad m \in {Npos} .  {{{{m}*{x}}={n}}\unicdbdjdcd{q}}}}})}\unicdbdjdcd{q}}}})}\unicdbdjdcd{{({\unicdcdadad x \in {R} .  {\unicdcdadad n \in {Z} .  {\unicdcdadad m \in {Npos} .  {{{{m}*{x}}={n}}\unicdbdjdcd{{x} \in {Q}}}}}})}\unicdbdjdcd{p}}}})}\unicdbdjdcd{p}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & df79085a1a2f4f031815dcd379e51112884d05ae157fcb74a8357af5dbe034eb \\
Pure Prop Address: & TMHAByNoEuG6wgDbba13PkCr4En34QBgFU2 \\
Theory Prop Id: & 61a4917a561c1dba953134725332117c4984e25d8b4082687743fbe846b3ddbb \\
Theory Prop Address: & TMFXPuxaie6AGgwAYXkFdSq3ph8iYboGfCE
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfbfpfcf} ${{Q} \subseteq {R}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b58f7fc652d381159596cca7859b3e14f253ce70a6c3e82c028fd369e27be76d \\
Pure Prop Address: & TMPV2a7aaKXBnCmnZWeiVvEJgyUM2fJ74rB \\
Theory Prop Id: & 2ff2bd5001c8ea913870f94bda57e521ab15c3861a373ea631397059ee56519a \\
Theory Prop Address: & TMExUYP3PJpDU1qNcRiaxE1KgojHSA3GBtE
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfkfpfbf} ${{Z} \subseteq {Q}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c1725ba9fa1a646ac94926c0e35963caee38e306b5ea5c0b8035f91f3edf3415 \\
Pure Prop Address: & TMZAUPnvZW22Vu1BjruFvkzzd99KRdyCUdr \\
Theory Prop Id: & a45e948eb32abcb85b27b170a391ad1b87a9608ce8a8b0a2ca2474cc443dc97e \\
Theory Prop Address: & TMNdjiF5RG9zcoL2KHoDujJUXF1wJoxRKBu
\end{tabular}
\end{center}

\end{theorem}
{\bf{Notation.}} We use $\infkdpckd$ as an infix operator  corresponding to applying term ${{\mname{Field\_div}}~{({{\mname{Field\_of\_RealsStruct}}~{Rs}})}}$.
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfegjgghpfdgmgpgdh} ${\unicdcdadad x \in {R} .  {\unicdcdadad y \in {{R}\unicdcdbdgd{\{{zero}\}}} .  {{{x}:/:{y}} \in {R}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ec4148787b44fb5fb3cdf6e61529dd3330a1e03298e8bbb7f08346f738f7db84 \\
Pure Prop Address: & TMJcMfeW6RzhudXxfGUAHHUfTerg3RCmgMh \\
Theory Prop Id: & d6d8aa472b7b2bd90954c9998de9b5a34d99b67efd39be56f8b6218f52cd2666 \\
Theory Prop Address: & TMJZqMkKWwmAcVqoAbsN2xJM74ofu6EF33P
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfngfhmgehpfegjggh} ${\unicdcdadad x \in {R} .  {\unicdcdadad y \in {{R}\unicdcdbdgd{\{{zero}\}}} .  {{x}={{y}*{({{x}:/:{y}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e2c1b9283ef45d7416d60f2ca96fe1d920af9c0d51788f43831368a679fa3ada \\
Pure Prop Address: & TMX4EqbemoaxGX4DUARDb4VZRc5ZLJMnF3R \\
Theory Prop Id: & b7ef46712de5c007a18453e0201b1b57274c06e0a53cd5a6ca5b05df985c4296 \\
Theory Prop Address: & TMds3wgEHVh1PGSrKhMkyaYaGRFs33AcAF5
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfegjgghpffhogegfgggbd} ${\unicdcdadad x y .  {{{x}\unicdcdadjd{R}}\unicdbdjdcd{{{x}:/:{y}}={0}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6dfdbeeddd4c9724f299edd163a3a6873b04737b480b6dc8622825fe39c1c4dc \\
Pure Prop Address: & TMc3no3gvLt4aCZ3tswEMYfjDATjdvWyatp \\
Theory Prop Id: & 351a0c5a15c5c52ea191c34690d189cbd566f2471f004c24cbe5973f3f13e435 \\
Theory Prop Address: & TMKJ4U4k2tPg572kNK69NYJYjNPh6PL3NV7
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfegjgghpffhogegfgggcd} ${\unicdcdadad x y .  {{{y}\unicdcdadjd{R}}\unicdbdjdcd{{{x}:/:{y}}={0}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f6f4785d660ace5156add885c383a28b7bc3a99a73d8631604e4197e578adf19 \\
Pure Prop Address: & TMWAnh5soQNHhi3Ykqn6R2ibResPkSHppAi \\
Theory Prop Id: & 38a3ac5076648eec3d5a9c2e50962482b5cbee9811940dc2427c63819c1ca1f1 \\
Theory Prop Address: & TMH1mYFWp41cUUu5C87Kq6Db727u6UDJH3K
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfegjgghpffhogegfgggdd} ${\unicdcdadad x .  {{{x}:/:{zero}}={0}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 36b6fd6a7618c15898b200e117ddef4e3fe1f0afb35443199399f43ed8036266 \\
Pure Prop Address: & TMch6Tu6CKzcYBgw8HNboBazjtiajd7yott \\
Theory Prop Id: & 269a3881564868f7961f0c10bad584f0aaaf8295d8a8d2960f2f7fd1e55d6fd8 \\
Theory Prop Address: & TMM4gQM96yMEzmMaUmB5ZXKT5bDGqjMZCBZ
\end{tabular}
\end{center}

\end{theorem}
{\bf{Notation.}} We use $\infof$ as a right associative infix operator  corresponding to applying term ${{\mname{CRing\_with\_id\_omega\_exp}}~{({{\mname{Field\_of\_RealsStruct}}~{Rs}})}}$.
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpgngfghgbgpffgihahpfad} ${\unicdcdadad x .  {{{x}^{0}}={one}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4f6df1679e5256f3dee7aa33e1d3edf948a982563cef834884ba4637ff642cfd \\
Pure Prop Address: & TMPSGSuE8LXDV4V4pvTtsJEu5FNyH1pBH5q \\
Theory Prop Id: & e9266c9ac7734a0908ed30b0412c09ac2fdda52a8a7deb8bfa1f329e7fb57079 \\
Theory Prop Address: & TMTnnud9giEgqcthVyEpMdJL9R2n8gzrP43
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpgngfghgbgpffgihahpfdf} ${\unicdcdadad x .  {\unicdcdadad n \in {\mname{omega}} .  {{{x}^{({{\mname{ordsucc}}~{n}})}}={{x}*{{x}^{n}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 02959608ed47b98e7731a645df3547cd11df919eee7b8db3daea36611099ac2b \\
Pure Prop Address: & TMVd93sgsKDaad1CxNYVmDVHrLcidvdTG4V \\
Theory Prop Id: & f566c18daaed324edc80d61391474e1103d351c36f7ff173e024f2c1aafb71d1 \\
Theory Prop Address: & TMPtaovCUa2PSiq9AjqGhtJZYWsZ7ChGjWW
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpgngfghgbgpffgihahpfbd} ${\unicdcdadad x \in {R} .  {{{x}^{1}}={x}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 68407b69e24d974f37306277e9073efc244d1ef9140fa0f4a8f4dd184e3a5a2a \\
Pure Prop Address: & TMSx7vspXi4hq9Zfc3jM1CSRCdp3Zc9NGRU \\
Theory Prop Id: & 0c96c808e943389302e04e16c0d612284ef50232f566ab8f9ee19bc68a0f29ee \\
Theory Prop Address: & TMKrzNRVaVw4oMu8nZ96vwqWBiBXmMPztXd
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpgngfghgbgpffgihahpfdgmgpgdh} ${\unicdcdadad x \in {R} .  {\unicdcdadad n \in {\mname{omega}} .  {{{x}^{n}} \in {R}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & eddf030a23ad3950942cdc2fbcd92ac5593fa9938ea1aae2da771538223f24df \\
Pure Prop Address: & TMNd3EMECuhQHTRvrFDgzibwvZjig6hvURt \\
Theory Prop Id: & ce17ebf726a60585318ead733cb7b5ab80320f5ea8b27994da5e5e3d5ce67456 \\
Theory Prop Address: & TMYuofpdnGFN8PpDrLruYivMbNZMsjVDEPo
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfbgcgdh}
$\mname{RealsStruct\_abs}$ is the opaque object of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & fcf3391eeb9d6b26a8b41a73f701da5e27e74021d7a3ec1b11a360045a5f13ca \\
Pure Object Address: & TMMiiibxSu8KWC8MJw2qM5UvvwMNYkqgGwE \\
Theory Object Id: & 480d2ba6a3a558d8d5642b2e68a88cdc65a67cf6bfe18dd0574c2b3d39d69d05 \\
Theory Object Address: & TMS4FH6cQhUENopB1oi9sMyEapmb4SckEPg
\end{tabular}
\end{center}

\end{definition}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfbgcgdhpfdgmgpgdh} ${\unicdcdadad x \in {R} .  {{{\mname{RealsStruct\_abs}}~{x}} \in {R}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 616c66d879b5e2f3a98bb920d8521cc473629f43225754cc970c502c02b4d70b \\
Pure Prop Address: & TMUJhPP2L2evvXGruzGDgRSyLBqRLkJsvew \\
Theory Prop Id: & 5a1facaf2b22cf8313c47ab5d00dd1d44a7f68dcc498ee31e588070e06e5ee9a \\
Theory Prop Address: & TMPi25yLHBrYYUawSVuNTAdtuM1J4ztYFgp
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfbgcgdhpfogpgogogfghgpfdgbgdhfg} ${\unicdcdadad x \in {R} .  {{{zero}\unicdcdgded{x}}\unicdbdjdcd{{{\mname{RealsStruct\_abs}}~{x}}={x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 883e07037d0bc45c4917edc7b4cb14370ef1994f68ceb7ca887b0471e7efe2ce \\
Pure Prop Address: & TMMrHyd4NfDirtPZFyor6Lq39dp81qYpayF \\
Theory Prop Id: & 6d71adbe5160105d6dc17d08068cceebbdc2513ee44ef22f79096d4f8800b735 \\
Theory Prop Address: & TMcT4sXVUoiheLTy9BFaCAh8cxDxrU7E51w
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfbgcgdhpfogfghgpfdgbgdhfg} ${\unicdcdadad x \in {R} .  {{{x}<{zero}}\unicdbdjdcd{{{\mname{RealsStruct\_abs}}~{x}}={\prenc{{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e8df199bfa3379375d084412f37f48b94a51f0ab84bb93db7eee356000d6afef \\
Pure Prop Address: & TMF7zwqh96P1YidqPVMDC2qQyYrNC6tSu2M \\
Theory Prop Id: & dc229d0f2268fec24b49f688c19646ae7a2e4a681cd5a1de6e740b17385573ba \\
Theory Prop Address: & TMFDA2267ptXsetvPjSP9B3kKZxjxiVe5eW
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfbgcgdhpfogpgogogfghg} ${\unicdcdadad x \in {R} .  {{zero}\unicdcdgded{{\mname{RealsStruct\_abs}}~{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3bae2c2e1b39e5830ba31107cc5578a7b625b83fec87496a6ae294027caaff03 \\
Pure Prop Address: & TMFabfpzFQBWazTGbdRKSFsh8H6hm1abtqe \\
Theory Prop Id: & 701a7053ca066eed86f0442240fc617b8c699f3d2e403833bad55bfc42fbe7db \\
Theory Prop Address: & TMK2MLY5rofcEXZj5urmPDkpnsWbeYS4r1w
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfbgcgdhpfkhfgchpgpfjgoggh} ${\unicdcdadad x \in {R} .  {{{{\mname{RealsStruct\_abs}}~{x}}={zero}}\unicdbdjdcd{{x}={zero}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1c1613207ea57ba5de38eb117ab32b134385a248879098c9d978b0898f59be2f \\
Pure Prop Address: & TMaZstUdzVeUwd6dG1Lk3yajQ9jPpSDmhtx \\
Theory Prop Id: & 480222f2b6e6e15591c55f3c22675fc071131e50cca95468ad1137f79bfc20cc \\
Theory Prop Address: & TMSwQ4V24ucAD2y6Y2NzBFzWw24QhLA9R2Q
\end{tabular}
\end{center}

\end{theorem}
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfegjgdhehpfkhfgchpgpffgbh} ${\unicdcdadad x y \in {R} .  {{{{\mname{RealsStruct\_abs}}~{({{x}+{\prenc{{y}}}})}}={zero}}\unicdbdjdcd{{x}={y}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0342d3a1b4534f4505f3b6ec468772cbfcd3cabf927b470703dcc7905876e881 \\
Pure Prop Address: & TMSygnAwf3Z1Nmyt9mu1mUiAZNnmpc8dwZS \\
Theory Prop Id: & d65f00cc24e6beb44968f1f8cd4eca4f2c99cc6a6e3fad31d5ee330fc5cba95b \\
Theory Prop Address: & TMG37mDdnRwQ9CZaHY6Sj3WmgU6SMiTvErN
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfegjgghjgegfgdh}
We define $\mname{RealsStruct\_divides}$ to be ${\uniddcgcg m n .  {\unicdcdaddd k \in {Npos} .  {{{m}*{k}}={n}}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 9b1a146e04de814d3707f41de0e21bf469d07ef4ce11ae2297f43acd3cb43219 \\
Pure Object Address: & TMSgvVAsMPWzeya7w14QXMLpm3oR2sg2Eaw \\
Theory Object Id: & 26b0fdc348372e6f151cfb0d9a85fba282630eaa98d67d43c3e14d87ae4d7907 \\
Theory Object Address: & TMEjvmG5htm9fo5j3Pi2G5irTZNBxF7JyLe
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfafchjgngfgdh}
We define $\mname{RealsStruct\_Primes}$ to be ${\{n \in {N}|{{{one}<{n}}\unicdcdcdhd{\unicdcdadad m \in {Npos} .  {{{{\mname{RealsStruct\_divides}}~{m}}~{n}}\unicdbdjdcd{{{m}={one}}\unicdcdcdid{{m}={n}}}}}}\}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 96c4c45b8c7f933c096a5643362b6a5eeb1d9e4d338d9854d4921130e6c8cdbe \\
Pure Object Address: & TMZiAsgfRmCK6xaRVxCPcc2VzGp7DAXMkTD \\
Theory Object Id: & 4f229104c32c2be9f21a21d40bf2c8848e077881ee9d18ea39563dd5055f3409 \\
Theory Object Address: & TMRZKCzNYB8mn6YTzXbD5KApbcoYvpYmMSv
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfdgpgahchjgngfg}
We define $\mname{RealsStruct\_coprime}$ to be ${\uniddcgcg m n .  {\unicdcdadad k \in {Npos} .  {{{{\mname{RealsStruct\_divides}}~{k}}~{m}}\unicdbdjdcd{{{{\mname{RealsStruct\_divides}}~{k}}~{n}}\unicdbdjdcd{{k}={one}}}}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 4857fdedbdc33f33b2c088479c67bbbceae1afd39fca032cb085f9f741cedca6 \\
Pure Object Address: & TMbzphBsd7GLnrNR8Mi8GuGapc1Xgdxr2y6 \\
Theory Object Id: & 112ea9cf8a89d5ad797d290cd1c0b26c9e1d6d41ecead75d5c7c7c91bd7ae48a \\
Theory Object Address: & TMX9ZHV8iS7puN2vhGXqUyRwbsJ5YunXJLK
\end{tabular}
\end{center}

\end{definition}
Let $Qs$ be ${{{{{{\mname{pack\_b\_b\_e\_e}}~{Q}}~{({{\mname{RealsStruct\_plus}}~{Rs}})}}~{({{\mname{RealsStruct\_mult}}~{Rs}})}}~{zero}}~{one}}$.
\begin{theorem}\label{thm:gejgfgmgegpfcffgbgmgdhdfehchfhdgehpfbf} ${{\mname{Field}}~{Qs}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8b600c18b435f30873008e1fd2e1a7754844b34c5ce89afb02dc5257a7d031f0 \\
Pure Prop Address: & TMLDM8xjNHCEb7tkdmX1V9EKpmqGB7zvxKT \\
Theory Prop Id: & b55584754578f3ef26cef3b37955ee96ee21f6de68eca6b9d14e62a3e2800837 \\
Theory Prop Address: & TMH6n1ZFLmtKDSDJVw8XnR3Do4UFL4NSg1Q
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:cffgbgmgdhdfehchfhdgehpfpgngfghgbgpffgngcgfgegegjgoghg}
We define $\mname{RealsStruct\_omega\_embedding}$ to be ${{{\mname{nat\_primrec}}~{zero}}~{({\uniddcgcg \_ r .  {{r}+{one}}})}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 7e897f0ce092b0138ddab8aa6a2fc3352b0c70d9383c22f99340c202779c7a39 \\
Pure Object Address: & TMFS8k2m5BMHnDtneuBVexHr4VeDPJZVm2S \\
Theory Object Id: & 8d1b1d11f8d7a646ada866cb891c4f97971a895e92e965d05e6be6a0bdc0a0ed \\
Theory Object Address: & TMLEHo8yiHeRBmSAVenhHZym5uhudBeWgVA
\end{tabular}
\end{center}

\end{definition}
Let $emb : {{\iota}\unicdbdjdcd{\iota}}$ be ${\mname{RealsStruct\_omega\_embedding}}$.
\begin{theorem}\label{thm:cffgbgmgdhdfehchfhdgehpfpgngfghgbgpffgngcgfgegegjgoghgpfoe} ${\unicdcdadad n \in {\mname{omega}} .  {{{emb}~{n}} \in {N}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 25cb31475191d8d41f82ac1cc047f3940af8421d025ff7a7c78fb4dc2ffe1cbe \\
Pure Prop Address: & TMPtZMNGEv6XXfgBKcBe8kTdZ3eDJkq7yCv \\
Theory Prop Id: & 9c35b850fb3d8b0adb59d8c3695495c52fc50505407deb7cccb631635d035d6a \\
Theory Prop Address: & TMXJrPBAAoNfUBWRhgmzF7gfY3kLs2rzR2q
\end{tabular}
\end{center}

\end{theorem}
