\chapter{Surreal Numbers I}
\section{Surreal Numbers as Predicates on Ordinals} % TaggedSets
\begin{definition}\label{def:afoepgfebhpf}
We define $\mname{PNoEq\_}$ to be ${\uniddcgcg \alpha p q .  {\unicdcdadad \beta \in {\alpha} .  {{{p}~{\beta}}\unicdbdjded{{q}~{\beta}}}}}$ of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & d7d95919a06c44c69c724884cd5a474ea8f909ef85aae19ffe4a0ce816fa65fd \\
Pure Object Address: & TMWJ8MJ9pcvgRi55Dv3T4ZDyt4PLqinJMsL \\
Theory Object Id: & 144b6539b9fe7bd5fe6597bae5a8aafdcf9b0b0be449f56a38d2ec86b003c9fa \\
Theory Object Address: & TMFreHQ2BuPpMHdDn38YhqZKp88uZF4gkiP
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoEq_ref_
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfchfgggpf} ${\unicdcdadad \alpha .  {\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{p}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c27ebb17a66d8faafb90c6e2c087edacd9723fae48be3b2096fcbfb163cae3ba \\
Pure Prop Address: & TMR5Xh2p5AFCBWLLpLnHCLb42vf2t6zTXDM \\
Theory Prop Id: & df4cf66649d1fd1d26f29aadca2801400f850f3b4eafdc3be31d69bde8bff283 \\
Theory Prop Address: & TMVv8oi3PpMpj4sDpqjfj6YKhg5rdMeRTKn
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoEq_sym_
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfdhjhngpf} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{p}}}}~{{q}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{q}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b813f3d183ae4292b34de6ea5ae07acbc9fd890e0ca37ba1d70d462219b6b8bc \\
Pure Prop Address: & TMdRTKSCoAPR1yxWCt8NqSNPsCeHURA3SC1 \\
Theory Prop Id: & bfda0b2058ffa91747bd1ac3030674a6cc496315b46c8308434fba9aea3f2657 \\
Theory Prop Address: & TMLYYwg1Kxo3WZ71E4yBpKXxAE1s3vDazeX
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoEq_tra_
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfehchbgpf} $${\unicdcdadad \alpha .  {\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}\unicdbdjdcd{{{{{\mname{PNoEq\_}}~{\alpha}}~{q}}~{r}}\unicdbdjdcd{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{r}}}}}}.$$ The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & dded4c9b3c97511b828af400470ae4f1a1e6f19deb142324786a79e7a5e93f13 \\
Pure Prop Address: & TMHc7CPnyGLonvb5EfKN1tAwcVMJrF3RQEa \\
Theory Prop Id: & fc92f3d0615e8bb0c739baceeb42a209768755f252f7c3eec934cebe529e6569 \\
Theory Prop Address: & TMdXdRjndh2ZZ4bnALH7m92PyqRv7hAdFp5
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoEq_antimon_
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfbgogehjgngpgogpf} $${\unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  {\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad \beta \in {\alpha} .  {{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}\unicdbdjdcd{{{{\mname{PNoEq\_}}~{\beta}}~{p}}~{q}}}}}}}.$$ The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8868e6589767bf4a8cd7788af6bd77faeab57b18ffdcecbe07aba3ed7624aee9 \\
Pure Prop Address: & TMT6Y9Nfrocaqa6thWmudf8Un3qLyLduStm \\
Theory Prop Id: & 094ae21814e156023eeffb5f103f193c98ceca6be42e5d08f1b636e235fc167a \\
Theory Prop Address: & TMcxE6t2A2eCrodDrECpNG7uGeMtfeS2zeN
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgmeehpf}
  We define $\mname{PNoLt\_}$ to be
  $${\uniddcgcg \alpha p q .  {\unicdcdaddd \beta \in {\alpha} .  {{{{{{\mname{PNoEq\_}}~{\beta}}~{p}}~{q}}\unicdcdcdhd{\preoh{{{p}~{\beta}}}}}\unicdcdcdhd{{q}~{\beta}}}}}$$
  of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}$
  identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 34de6890338f9d9b31248bc435dc2a49c6bfcd181e0b5e3a42fbb5ad769ab00d \\
Pure Object Address: & TMZD4o1Ego5dxANfVMw5J8RL7t5naNG9U6L \\
Theory Object Id: & c95964975192a20b94bb694b4b9a7c55de7fce1d4a56cd9d0ed8d67ae58fb60e \\
Theory Object Address: & TMG3oBNXZHp4KrZAdKCT2kf3DvDUqGNax9b
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoLt_E_
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpffepf}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha .  \unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  {{{{\mname{PNoLt\_}}~{\alpha}}~{p}}~{q}}\unicdbdjdcd\unicdcdadad R : {o} . \\ {{({\unicdcdadad \beta .  {{{\beta} \in {\alpha}}\unicdbdjdcd{{{{{\mname{PNoEq\_}}~{\beta}}~{p}}~{q}}\unicdbdjdcd{{\preoh{{{p}~{\beta}}}}\unicdbdjdcd{{{q}~{\beta}}\unicdbdjdcd{R}}}}}})}\unicdbdjdcd{R}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9ba5433aba8ff79cf5b29265ee6f028e168d178699edf447c85bbc8deeac2a4e \\
Pure Prop Address: & TMPQB13m96524sAUWzy2GhLxsUb6akgizyg \\
Theory Prop Id: & 45aec844f872697c4972ac0ebc6a05b529ce6188709968bbc3161616d02ce8ae \\
Theory Prop Address: & TMNravm4aL128MRxYXGMQuCHaauCV7HVCoo
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLt_irref_
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpfjgchchfgggpf} ${\unicdcdadad \alpha .  {\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {\preoh{{{{{\mname{PNoLt\_}}~{\alpha}}~{p}}~{p}}}}}}.$ The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fcb502c9cc3805a7a66fa875754cf8bf23ca05874f5dbdc9fa408a45bf6b32cd \\
Pure Prop Address: & TMQ9CemPxrwhC7R2QenNg6KpFZuCjjKb8x2 \\
Theory Prop Id: & 987dff18ff1b92d203fd612706b6d76ae7437c60c295931c75665e68331e9fff \\
Theory Prop Address: & TMYjm5nDdDXhStiue7yDo3jqogooZAR36uB
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLt_mon_
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpfngpgogpf}
  $$
    \unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  \unicdcdadad \alpha .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd\unicdcdadad \beta \in {\alpha} .  {{{{{\mname{PNoLt\_}}~{\beta}}~{p}}~{q}}\unicdbdjdcd{{{{\mname{PNoLt\_}}~{\alpha}}~{p}}~{q}}}.
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b1c3cf515eec97e9e1b0860995940355d51e18df35bc3f66115f683ebee76539 \\
Pure Prop Address: & TMJDSNwcnhwkzoXr3Z8kRHFuHXmWJPaX3Ce \\
Theory Prop Id: & d8d0f010ecea2f707f8d719d67b8d74d77b42bb87c9317a35e1a8a3e812312c5 \\
Theory Prop Address: & TMdFZnNNsv6frZTGhJdUhF23hzzvniYV8hv
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLt_trichotomy_or_
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpfehchjgdgigpgehpgngjhpfpgchpf} $${\unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  {\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{{{{\mname{PNoLt\_}}~{\alpha}}~{p}}~{q}}\unicdcdcdid{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}}\unicdcdcdid{{{{\mname{PNoLt\_}}~{\alpha}}~{q}}~{p}}}}}}.$$ The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6644084c8b8082d13eb4f918c0458a6a57e0e4c038c596a1395d3eb99cd50e79 \\
Pure Prop Address: & TMcyZEERpexRUZhwpuM9GSuX3Sk5rPkRLNj \\
Theory Prop Id: & 8d3824c21ec2e23224adc53da96078403c04f5984cb1b5363e6033afb3557ba8 \\
Theory Prop Address: & TMdzU9ADKUux6zhuXTRcw9NBL5GrWiZFnL5
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLt_tra_
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpfehchbgpf} $${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{\mname{PNoLt\_}}~{\alpha}}~{p}}~{q}}\unicdbdjdcd{{{{{\mname{PNoLt\_}}~{\alpha}}~{q}}~{r}}\unicdbdjdcd{{{{\mname{PNoLt\_}}~{\alpha}}~{p}}~{r}}}}}}}.$$ The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3d7d91c93d4e62413c6d9dd981bb8b73fe7cceb7d203f0798d848121d5071de2 \\
Pure Prop Address: & TMLwvVmBwmxcKpTr11BfuohRJkbqDNeYgkK \\
Theory Prop Id: & 44767a86f732b5f6b0f1ea4f2fd902fb4c657cd5b93d818dccf9adaf334006e4 \\
Theory Prop Address: & TMXUBoU9EyL1j4brgyEDkPSkDKW8vBUKaAk
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgmeeh}
  $\mname{PNoLt}$ is the opaque object of type
  $${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$$
  identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 8f57a05ce4764eff8bc94b278352b6755f1a46566cd7220a5488a4a595a47189 \\
Pure Object Address: & TMQBApBYGx5gde2jLgpjV5KGX8BPSrspNY9 \\
Theory Object Id: & f2094ca61fcaf27eecff3b6b119f415eaf83537511fa1030a24f9fa1658338a9 \\
Theory Object Address: & TMUcNNa6THZvpai86WFEVkAxVdKBFzgWMLt
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoLtI1
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehjebd} 
$$
\begin{array}{c}
{\unicdcdadad \alpha \beta .  }{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{\mname{{PNoLt\_}}}}~{{({{{{\alpha}}\unicdcdcdjd{{\beta}}}})}}}}~{{p}}}}~{{q}}}}}{\unicdbdjdcd}{{{{{{{{{{\mname{{PNoLt}}}}~{{\alpha}}}}~{{p}}}}~{{\beta}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c5735c7eb7f3308228886e03ec77196c4e828c0817e59a83eb6ea91b420edb3e \\
Pure Prop Address: & TMS9TizXPog5PcwVnGnsKDk5q9DPz7TbhoH \\
Theory Prop Id: & ecbc7189a71dd70627c550c8d9fdc53c6bd779fdc38db26b5b96dd819de399c8 \\
Theory Prop Address: & TMdBwxsaeBCn1Cz8duce8N385NuVba9M1iz
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLtI2
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehjecd} $${\unicdcdadad \alpha \beta .  {\unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  {{{\alpha} \in {\beta}}\unicdbdjdcd{{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}\unicdbdjdcd{{{q}~{\alpha}}\unicdbdjdcd{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}}}}}}.$$ The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 419b53c5b7f090cd6916ecca9011b3d8d19ae95bc60020cad45d8c92803c2005 \\
Pure Prop Address: & TMLoQH9MBcB6zQZDo5PkbRneUZBXQQRzDr8 \\
Theory Prop Id: & 030093820795845274dac213f6f565752cfe3de0e2652c9c76ca161494618286 \\
Theory Prop Address: & TMGXNx56uxbqYFaCe1PxneJjhYYGcLCyE8T
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLtI3
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehjedd} $${\unicdcdadad \alpha \beta .  {\unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  {{{\beta} \in {\alpha}}\unicdbdjdcd{{{{{\mname{PNoEq\_}}~{\beta}}~{p}}~{q}}\unicdbdjdcd{{\preoh{{{p}~{\beta}}}}\unicdbdjdcd{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}}}}}}.$$ The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c392cb53f2ad2f2d109882679fc4b8a23c27eeb0ff9a5fef31347a885ba891da \\
Pure Prop Address: & TMNZw7Cj3j1w5e5hoebGCVirnCRB3RXaNvW \\
Theory Prop Id: & 4104429992b8f35592b8ceba357f6dc0dc7f2ba8a5f6b06451be23cdd6fb161b \\
Theory Prop Address: & TMFhY7wPRcF4NknzzkXMPbsPrBbMZ2PZQb8
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLtE
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehfe}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta .  \unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  {{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdbdjdcd\unicdcdadad R : {o} .\\  {({{{{{\mname{PNoLt\_}}~{({{\alpha}\unicdcdcdjd{\beta}})}}~{p}}~{q}}\unicdbdjdcd{R}})}\unicdbdjdcd\\
      {({{{\alpha} \in {\beta}}\unicdbdjdcd{{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}\unicdbdjdcd{{{q}~{\alpha}}\unicdbdjdcd{R}}}})}\unicdbdjdcd\\
        {{({{{\beta} \in {\alpha}}\unicdbdjdcd{{{{{\mname{PNoEq\_}}~{\beta}}~{p}}~{q}}\unicdbdjdcd{{\preoh{{{p}~{\beta}}}}\unicdbdjdcd{R}}}})}\unicdbdjdcd{R}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f7e6e3e5f56759996f8e71b71ecc1cfeba2385a738e3535fb9f45c5f05a7ba7a \\
Pure Prop Address: & TMPLU4MZci1RZyu9YajBNzBvAn2625YmMhz \\
Theory Prop Id: & af36ceaf250752d7f1c85964b80e0b0df1acb7a0538c6e38f8cd6b431c530c5e \\
Theory Prop Address: & TMbpk33jSr7sb7xpz9amLj1PFYcwRPKWBzJ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLtE2
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehfecd} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{{{\mname{{PNoLt}}}}~{{\alpha}}}}~{{p}}}}~{{\alpha}}}}~{{q}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{PNoLt\_}}}}~{{\alpha}}}}~{{p}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ad392c5c3af8700314707c033e9816397448a5fcd60aab607c4fa7af5a143745 \\
Pure Prop Address: & TMX9jn8QF6K3SxcZrS4opjiQZ9yxZ4mP3ju \\
Theory Prop Id: & 7ef10485fe210dd748b752318940f7a2ccbe8a44d685b1525c931e391a9d5662 \\
Theory Prop Address: & TMX4uzbNMJMGDv9jTKuePDdqiWegaum41fi
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLt_irref
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpfjgchchfggg} ${\unicdcdadad \alpha .  {\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {\preoh{{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\alpha}}~{p}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ff1a80a33f6f089b5a40873efe47b5c845fd624b4250c5c7bb6d8c4de6bc398d \\
Pure Prop Address: & TMYgpDGCHGRHDBZdPr12n2a5kcKFbuHdceA \\
Theory Prop Id: & d4c3797f7ec820194be1429644bfb16565fc442db2d2480bac78b9f481bb2a97 \\
Theory Prop Address: & TMdk8u8CYrFAFVf8jmy3pdMCDAMd2SPuATi
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLt_trichotomy_or
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpfehchjgdgigpgehpgngjhpfpgch}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta .  \unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd\\
                 {{{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdcdcdid{{{\alpha}={\beta}}\unicdcdcdhd{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}}}\unicdcdcdid{{{{{\mname{PNoLt}}~{\beta}}~{q}}~{\alpha}}~{p}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4f39e49b96938924ecee0c62b18b8054abc06386845a55af32ecbc5c07a60f7d \\
Pure Prop Address: & TMcxeDW7jxsMf6iBBZyA1rMm28N9LGxrxAe \\
Theory Prop Id: & 90b38722517f98c1ebbb09338d5ac216a2660c649969e817ce4af46864ab1174 \\
Theory Prop Address: & TMKtLGK85iJ16Tv3NbRNth1Uc13sd3VWcEC
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLtEq_tra
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehfebhpfehchbg}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd\\
          {\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdbdjdcd{{{{{\mname{PNoEq\_}}~{\beta}}~{q}}~{r}}\unicdbdjdcd{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{r}}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2e082ab6742a9e96182b14bc401ef1691287ea931ad64b2d9f8143dd38e72381 \\
Pure Prop Address: & TMKVFaHQYU7RMkic7ehXR1Hz5AxwJLND1Xs \\
Theory Prop Id: & 2f2a828f72a2bc454fe6d454ef8a36f6aad539b02f33d5546be1b804baa1f49b \\
Theory Prop Address: & TMVam3H7JTsW4GptAT9oJpfx8ncfn8EV3XV
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoEqLt_tra
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhmeehpfehchbg}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd\\
                 {\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}\unicdbdjdcd{{{{{{\mname{PNoLt}}~{\alpha}}~{q}}~{\beta}}~{r}}\unicdbdjdcd{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{r}}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fe144b0ff77f85ded00481bd97ae472212a28995d89fa3667ab8889dff197bb5 \\
Pure Prop Address: & TMVXYVUmzv2xFHz12tNLMKuyKAeKQiURGoM \\
Theory Prop Id: & f076541f250b71cb6126517531b079c63b10d79f85685776f781bf746959df68 \\
Theory Prop Address: & TMRsHUfrwX3yw5eXZcC3S8S9yJKYP2vG94Z
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLt_tra
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpfehchbg}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta \gamma .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{\mname{ordinal}}~{\gamma}}\unicdbdjdcd\\
                 {\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdbdjdcd{{{{{{\mname{PNoLt}}~{\beta}}~{q}}~{\gamma}}~{r}}\unicdbdjdcd{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\gamma}}~{r}}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b72127d429a420ed218c832bea2f3aaf8a09149da0276d6333614f557fff90a7 \\
Pure Prop Address: & TMV9P8LWZv2DWhZZm536XdWjUpfNXSoY83p \\
Theory Prop Id: & 8eff85e4785607b84b3c70850a18baea404f11fd03cc04cfd5afa1cb189a0c5f \\
Theory Prop Address: & TMGtkttdVbaT9NVAgziSNR2tFzasudosPWy
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgmefg}
We define $\mname{PNoLe}$ to be $${\uniddcgcg \alpha p \beta q .  {{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdcdcdid{{{\alpha}={\beta}}\unicdcdcdhd{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}}}}$$ of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & ac6311a95e065816a2b72527ee3a36906afc6b6afb8e67199361afdfc9fe02e2 \\
Pure Object Address: & TMa2eyC7pjLoLi2qJAssPFeBwpm49h4vYjE \\
Theory Object Id: & 770cf7219dc1ea3a0f1a555c8be6a16fbc83df72a88b338d5a9b6c5282933d86 \\
Theory Object Address: & TMHvhi4cZx4BpvWATQS4A5YGwuJKHFHnqZ8
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoLeI1
\end{verbatim}
\begin{theorem}\label{thm:afoepgmefgjebd} 
$$
\begin{array}{c}
{\unicdcdadad \alpha \beta .  }{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{{{\mname{{PNoLt}}}}~{{\alpha}}}}~{{p}}}}~{{\beta}}}}~{{q}}}}}{\unicdbdjdcd}{{{{{{{{{{\mname{{PNoLe}}}}~{{\alpha}}}}~{{p}}}}~{{\beta}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1f0528aa090a84e44d3bcd21d1f5c364665fee6e39da73097b72aa626f61a067 \\
Pure Prop Address: & TMMU61EKjZQ56vwqnq9EJcT3RZD4kTLXmEj \\
Theory Prop Id: & 674a2cb5c37edfb9cea0bddf54687b384d613f51ce354dd54fc543730f626cc9 \\
Theory Prop Address: & TMHGpziusAgRkcNhfxaVjHNxAiJnaP9Aq4m
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLeI2
\end{verbatim}
\begin{theorem}\label{thm:afoepgmefgjecd} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{p}}}}~{{q}}}}}{\unicdbdjdcd}{{{{{{{{{{\mname{{PNoLe}}}}~{{\alpha}}}}~{{p}}}}~{{\alpha}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b453329022d5a4973fd4979e8276c334bd9855b0fcef38a490d0d3832d17f295 \\
Pure Prop Address: & TMRnggtbHEorcGeqrrjsVRdcj87SoZCNfE9 \\
Theory Prop Id: & 0b2157420981c9fb11d49eda5ed224af6f8a3582031030525c97a2cb5d7f2065 \\
Theory Prop Address: & TMEuEJtVZKAPqoRvcrAuffF91gTB2F6p1ur
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLe_ref
\end{verbatim}
\begin{theorem}\label{thm:afoepgmefgpfchfggg} ${\unicdcdadad \alpha .  {\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{{{{\mname{PNoLe}}~{\alpha}}~{p}}~{\alpha}}~{p}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1b91bdb4dd7268a37d5bc57b8815bd8867df7f57b837c68a5a3cecba131ffe39 \\
Pure Prop Address: & TMZFLxhVzuJSDerZWEiaV9vKjwvZbajaqne \\
Theory Prop Id: & 613386d1779902b3b59aa4f008e13853ef42f333df28e697121c9469f82e8a8d \\
Theory Prop Address: & TMNTwN9qruHGfPxKb1SjuvQHFwhXyGnA1sj
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLe_antisym
\end{verbatim}
\begin{theorem}\label{thm:afoepgmefgpfbgogehjgdhjhng}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd\\
          {\unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  {{{{{{\mname{PNoLe}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdbdjdcd{{{{{{\mname{PNoLe}}~{\beta}}~{q}}~{\alpha}}~{p}}\unicdbdjdcd{{{\alpha}={\beta}}\unicdcdcdhd{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 934d7a99d54f73bf9a9e6317bc9decff59c4047034ee26980b50587b2dd95801 \\
Pure Prop Address: & TMNTPG6YJEwCxRs1V24gTDCmTzHzRvRc6gk \\
Theory Prop Id: & 4e35944f68e317a6041fb50a04c326a498b0c1bc1e0ed7e8064b1a16614489aa \\
Theory Prop Address: & TMLEM4Wo7kALZKsWnk4Wd5KjwhRhR5uC1HT
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLtLe_tra
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehmefgpfehchbg}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta \gamma .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{\mname{ordinal}}~{\gamma}}\unicdbdjdcd\\
                 {\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdbdjdcd{{{{{{\mname{PNoLe}}~{\beta}}~{q}}~{\gamma}}~{r}}\unicdbdjdcd{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\gamma}}~{r}}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1565862dfbb792dc415a313525571261e4fc5597f958d0f6839b147404b16497 \\
Pure Prop Address: & TMS5q2iXwJmrbrvpyaPEgKeFc4yT8unvrtd \\
Theory Prop Id: & 09246b78705144b6b5679fa64a57f406c40b20b13b935e7a3c426c016425ec52 \\
Theory Prop Address: & TMP8YWTicgVUQEKtiP6hLWexaY1AtmqKF7Y
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLeLt_tra
\end{verbatim}
\begin{theorem}\label{thm:afoepgmefgmeehpfehchbg}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta \gamma .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{\mname{ordinal}}~{\gamma}}\unicdbdjdcd\\
                 {\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{{\mname{PNoLe}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdbdjdcd{{{{{{\mname{PNoLt}}~{\beta}}~{q}}~{\gamma}}~{r}}\unicdbdjdcd{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\gamma}}~{r}}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a260ca2dcf87c48e05a6f7916c83ca227958075cb681e56d4f97f095c7890142 \\
Pure Prop Address: & TMEugnvG1Bt7rUt9psKYMG4B3eXfMrHtrVq \\
Theory Prop Id: & ad6971ea2f07aa1305f0c6bc480a834ada5728150db9419ad7ccd521ebe1ce04 \\
Theory Prop Address: & TMT7tULGek9mdvm2GJBcURRtAYaQxeEmHmE
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoEqLe_tra
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhmefgpfehchbg}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd \\
                 {\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}\unicdbdjdcd{{{{{{\mname{PNoLe}}~{\alpha}}~{q}}~{\beta}}~{r}}\unicdbdjdcd{{{{{\mname{PNoLe}}~{\alpha}}~{p}}~{\beta}}~{r}}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7d3ff80df17ec96e504b4964272829064a97b36a1e75174ea67fe631b2aa92ab \\
Pure Prop Address: & TMWEkTHjED61EVaMYyFDrbsBT3uuPxsd2WS \\
Theory Prop Id: & 002ebf808b3010e1129d1a3d0f3318345316603ff5e9f8a4752963619b88ad30 \\
Theory Prop Address: & TMX3rf1dBjD7HFURU7toJLUuPtwumdeBSD4
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLeEq_tra
\end{verbatim}
\begin{theorem}\label{thm:afoepgmefgfebhpfehchbg}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd\\
                 {\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{{\mname{PNoLe}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdbdjdcd{{{{{\mname{PNoEq\_}}~{\beta}}~{q}}~{r}}\unicdbdjdcd{{{{{\mname{PNoLe}}~{\alpha}}~{p}}~{\beta}}~{r}}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2def03362104b85520032b762fc8d4c1d8df8e5ec52650b7c3b2e85cae9c2fcc \\
Pure Prop Address: & TMJmHGCtRPZGoKvdPGHCWSTaoKXedqvvXcT \\
Theory Prop Id: & ed30aa19eb8fe59fe9c61636687168d8be8ac512c5d98ff2581121ba1ffd0765 \\
Theory Prop Address: & TMRHBJiUqPVNK9rBz9VWXHRyyHjTuJD3RGU
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLe_tra
\end{verbatim}
\begin{theorem}\label{thm:afoepgmefgpfehchbg}
  $$
  \begin{array}{c}
    \unicdcdadad \alpha \beta \gamma .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{\mname{ordinal}}~{\gamma}}\unicdbdjdcd\\
                 {\unicdcdadad p q r : {{\iota}\unicdbdjdcd{o}} .  {{{{{{\mname{PNoLe}}~{\alpha}}~{p}}~{\beta}}~{q}}\unicdbdjdcd{{{{{{\mname{PNoLe}}~{\beta}}~{q}}~{\gamma}}~{r}}\unicdbdjdcd{{{{{\mname{PNoLe}}~{\alpha}}~{p}}~{\gamma}}~{r}}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 70b92b6662709f7dc5822fd97e53303a3a2d055ae004b41c4e645dce93c651bc \\
Pure Prop Address: & TMHzFcB1asSRVnHkPCarNFE3z9eXcp4jGKN \\
Theory Prop Id: & 4b557dc79b8f761dcdf6d3964db848fe773041a6fb41c452623328cec43d42d1 \\
Theory Prop Address: & TMRqHazfJDQBeT8YDQifbd4FBKaDUPSHefM
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgpfegpghhogdg}
We define $\mname{PNo\_downc}$ to be $${\uniddcgcg L \alpha p .  {\unicdcdaddd \beta .  {{{\mname{ordinal}}~{\beta}}\unicdcdcdhd{\unicdcdaddd q : {{\iota}\unicdbdjdcd{o}} .  {{{{L}~{\beta}}~{q}}\unicdcdcdhd{{{{{\mname{PNoLe}}~{\alpha}}~{p}}~{\beta}}~{q}}}}}}}$$ of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 93dab1759b565d776b57c189469214464808cc9addcaa28b2dbde0d0a447f94d \\
Pure Object Address: & TMFFjJ65YM9kJCjR5s8uzZawGiUXuQM1tjB \\
Theory Object Id: & f6c96279232d9e9cca23ffe49c0c29166fada636c1195de7ffbd7fabd07ac1c7 \\
Theory Object Address: & TMbMkrzAgHfL7RAFKvYtv8vLTv3CdoSjov9
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afoepgpffhahdg}
  We define $\mname{PNo\_upc}$ to be
  $${\uniddcgcg R \alpha p .  {\unicdcdaddd \beta .  {{{\mname{ordinal}}~{\beta}}\unicdcdcdhd{\unicdcdaddd q : {{\iota}\unicdbdjdcd{o}} .  {{{{R}~{\beta}}~{q}}\unicdcdcdhd{{{{{\mname{PNoLe}}~{\beta}}~{q}}~{\alpha}}~{p}}}}}}}$$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 5437127f164b188246374ad81d6b61421533c41b1be66692fb93fdb0130e8664 \\
Pure Object Address: & TMaAkmpSZzSfoMddDbQnVgH1TnDuodvBF4s \\
Theory Object Id: & b97935067aeb5829f0c939603d3342813777d67c23748ee242158f33ac42ed50 \\
Theory Object Address: & TMdz4JctWDDKNbEgwhwYLZAiTPcFhWgefwa
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoLe_downc
\end{verbatim}
\begin{theorem}\label{thm:afoepgmefgpfegpghhogdg}
  $$
  \begin{array}{c}
    \unicdcdadad L : {{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}} .  \unicdcdadad \alpha \beta .  \unicdcdadad p q : {{\iota}\unicdbdjdcd{o}} .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{ordinal}}~{\beta}}\unicdbdjdcd\\
              {{{{{\mname{PNo\_downc}}~{L}}~{\alpha}}~{p}}\unicdbdjdcd{{{{{{\mname{PNoLe}}~{\beta}}~{q}}~{\alpha}}~{p}}\unicdbdjdcd{{{{\mname{PNo\_downc}}~{L}}~{\beta}}~{q}}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6b1f0d64dd061a5eba7c0342d7f3676187349916ec76342278678c84b205b24f \\
Pure Prop Address: & TMWAyyBEGMhjikXgumQbCvdi5WRdMG4XswZ \\
Theory Prop Id: & cd130e6e61b7fe7f0bfe1dd262ffaa5f86165a48fa9f622a9e0d11503c799ff3 \\
Theory Prop Address: & TMM4WeADC271qKWhMRuk4GonoM9GLTxaXM9
\end{tabular}
\end{center}
\end{theorem}
\begin{verbatim}
PNo_downc_ref
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfegpghhogdgpfchfggg}
  $$
  \begin{array}{c}
    \unicdcdadad L : {{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}} .  \unicdcdadad \alpha .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} . \\
                 {{{L}~{\alpha}}~{p}}\unicdbdjdcd {{{{\mname{PNo\_downc}}~{L}}~{\alpha}}~{p}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 347aa9ec835b739b3314c47b28ca0cd2ebf6b08b10135cc7cb91d5cdf39bed56 \\
Pure Prop Address: & TMT5dDg6pVSTQmj3sYdZvHj975JFuQSEf3R \\
Theory Prop Id: & 69dfcb8516ec52d7595fe7c4aaf6805468aa6d2634d29ea2b8226ad567420a27 \\
Theory Prop Address: & TMUEbPAgDX5aYyR8xXjQWGAsxbj8SwH4QN4
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_upc_ref
\end{verbatim}
\begin{theorem}\label{thm:afoepgpffhahdgpfchfggg}
  $$
  \begin{array}{c}
    \unicdcdadad R : {{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}} .  \unicdcdadad \alpha .  {{\mname{ordinal}}~{\alpha}}\unicdbdjdcd\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .\\
                 {{{{R}~{\alpha}}~{p}}\unicdbdjdcd{{{{\mname{PNo\_upc}}~{R}}~{\alpha}}~{p}}}.
  \end{array}
  $$
  The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4039424bd88802176a6cbb56b76abcf8b8a1f616a3d0721d28e58f9f096aa816 \\
Pure Prop Address: & TMRis4sdu68KHLQb77r68nGDtFxyhKXKcBM \\
Theory Prop Id: & c4bf71e4c9e3609364a3a0c2ce3ae0a4befd6f3181cea4eec8656173a1b32b72 \\
Theory Prop Address: & TMFX2E2HrfER46NsW7mPNvJqqJZFcAuJx1i
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLe_upc
\end{verbatim}
\begin{theorem}\label{thm:afoepgmefgpffhahdg} 
$$
\begin{array}{c}
{\unicdcdadad R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha \beta .  }{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{\mname{{ordinal}}}}~{{\beta}}}}}{\unicdbdjdcd}\\
{{{{{{{{{\mname{{PNo\_upc}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}{{{{{{{{{{{\mname{{PNoLe}}}}~{{\alpha}}}}~{{p}}}}~{{\beta}}}}~{{q}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{PNo\_upc}}}}~{{R}}}}~{{\beta}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 111b74916dd3b9d72613111538f033f3bb8b5dfe2d40082b76d45a031efb7cc7 \\
Pure Prop Address: & TMcxmbHaZztM29ZiLWZpgrqw4nZNUzfJYeS \\
Theory Prop Id: & eb9ed5adca244da4ff7ae93d024a4823361cb4695e55f8fa465e7e14c460eae2 \\
Theory Prop Address: & TMJYddW3A9sz7atvMHDox6oK3Fz5ZXhVgw5
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgmeehpfahhhjgdhfg}
  We define $\mname{PNoLt\_pwise}$ to be
  $$
  \begin{array}{c}
    \uniddcgcg L R .\\
    \unicdcdadad \gamma .  {{\mname{ordinal}}~{\gamma}}\unicdbdjdcd\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{{L}~{\gamma}}~{p}}\unicdbdjdcd\\
    \unicdcdadad \delta .  {{\mname{ordinal}}~{\delta}}\unicdbdjdcd\unicdcdadad q : {{\iota}\unicdbdjdcd{o}} .  {{{R}~{\delta}}~{q}}\unicdbdjdcd\\
                 {{{{{\mname{PNoLt}}~{\gamma}}~{p}}~{\delta}}~{q}}
  \end{array}
  $$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & ae448bdfa570b11d4a656e4d758bc6703d7f329c6769a116440eb5cde2a99f6f \\
Pure Object Address: & TMc96g2JPpBeTmDVeFvvSZTGX5Tb5262YGw \\
Theory Object Id: & 12380893a7dd9882f429d48a7961599b1aa9e328a364d33120d9b2b3f2b04f70 \\
Theory Object Address: & TMQdLYLNu2LJgg1svc2g1R5FdDAS2iXTH2E
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoLt_pwise_downc_upc
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpfahhhjgdhfgpfegpghhogdgpffhahdg} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}\\
{{{{{{\mname{{PNoLt\_pwise}}}}~{{({{{{\mname{{PNo\_downc}}}}~{{L}}}})}}}}~{{({{{{\mname{{PNo\_upc}}}}~{{R}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2b8e3bd1c2f83410acc3eb16745b3c189b91f3d88eff48c21bc1fe2da62df2a9 \\
Pure Prop Address: & TMaupVE3uvrBXnhCNWJG4uidezoNueB9Pzn \\
Theory Prop Id: & 2f244c622c329868222671ac992535b2ec8a107cab98a55451d89fc062755198 \\
Theory Prop Address: & TMSZ221fidpj3oVVEjh7AxtKFSvU4SKPRwZ
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgpfchfgmgpfdhehchjgdgehpffhahahfgchcgeg}
  We define $\mname{PNo\_rel\_strict\_upperbd}$ to be
  $$
    \uniddcgcg L \alpha p .  \unicdcdadad \beta \in {\alpha} .  \unicdcdadad q : {{\iota}\unicdbdjdcd{o}} .  {{{{\mname{PNo\_downc}}~{L}}~{\beta}}~{q}}\unicdbdjdcd
               {{{{{\mname{PNoLt}}~{\beta}}~{q}}~{\alpha}}~{p}}
  $$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & dc1665d67ed9f420c2da68afc3e3be02ee1563d2525cbd6ab7412038cd80aaec \\
Pure Object Address: & TMPiEDqG64KxZbAF6RkpdrzFu2A2HmgQJ6p \\
Theory Object Id: & 3e1663abe9ff987d905d14ab1ac2db49036bac313db730a8f2e8fbb4db756c76 \\
Theory Object Address: & TMGu9LKc1Ew275sc36SoMWGV6V7yydJCxoa
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afoepgpfchfgmgpfdhehchjgdgehpfmgpghhfgchcgeg}
  We define $\mname{PNo\_rel\_strict\_lowerbd}$ to be
  $$
    \uniddcgcg R \alpha p .  \unicdcdadad \beta \in {\alpha} .  \unicdcdadad q : {{\iota}\unicdbdjdcd{o}} . {{{{\mname{PNo\_upc}}~{R}}~{\beta}}~{q}}\unicdbdjdcd
               {{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}
  $$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & e5a4b6b2a117c6031cd7d2972902a228ec0cc0ab2de468df11b22948daf56892 \\
Pure Object Address: & TMMtUywXSfQieSYmZXKhRaMVMvNBJ4mv5n1 \\
Theory Object Id: & ff3150c8fe3f147f482d147d4948a3adfa0f23616d03c877213e009957f55a8e \\
Theory Object Address: & TMUamWcr8Mykvg8ZyWAHjTGbXtZ7LdvyWix
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afoepgpfchfgmgpfdhehchjgdgehpfjgnggh}
  We define $\mname{PNo\_rel\_strict\_imv}$ to be
  $$
  \uniddcgcg L R \alpha p .  {{{{\mname{PNo\_rel\_strict\_upperbd}}~{L}}~{\alpha}}~{p}}\unicdcdcdhd{{{{\mname{PNo\_rel\_strict\_lowerbd}}~{R}}~{\alpha}}~{p}}
  $$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 64ce9962d0a17b3b694666ef1fda22a8c5635c61382ad7ae5a322890710bc5f8 \\
Pure Object Address: & TMFES6j6egEfn4hHem6CQT3DsNYsaxQYweU \\
Theory Object Id: & 5b35fbff3a2443e9ad06778c49bcebadc3d05d5d8c2da1fb0fad3c9047e95755 \\
Theory Object Address: & TMXkoajkux97senphT28NAy8HKer2u4kPUc
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoEq_rel_strict_upperbd
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfchfgmgpfdhehchjgdgehpffhahahfgchcgeg} 
$$
\begin{array}{c}
{\unicdcdadad L : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{p}}}}~{{q}}}}}{\unicdbdjdcd}\\
{{{{{{{{{\mname{{PNo\_rel\_strict\_upperbd}}}}~{{L}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{\mname{{PNo\_rel\_strict\_upperbd}}}}~{{L}}}}~{{\alpha}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & bb65cf435cd1184f19969d40225b30f85ec7629c2a8b3845828f9cae4b8218cd \\
Pure Prop Address: & TMbeVMWawJhuN1ZSdk7xfgpCpASxk2FoVZv \\
Theory Prop Id: & db4b8216b0bda52674a1b262e30b3b493a88c2284eb09e35892c6464171da698 \\
Theory Prop Address: & TMR2fFPGsUbreZFxGSKzs2Ut4aBjCbtG8FU
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_rel_strict_upperbd_antimon
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfchfgmgpfdhehchjgdgehpffhahahfgchcgegpfbgogehjgngpgog} 
$$
\begin{array}{c}
{\unicdcdadad L : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{\unicdcdadad \beta \in }{{{\alpha}}}{ .  }\\
{{{{{{{{{\mname{{PNo\_rel\_strict\_upperbd}}}}~{{L}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{\mname{{PNo\_rel\_strict\_upperbd}}}}~{{L}}}}~{{\beta}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9986afc591c8630816ae93586b1d747b9e6a7d43426e319ea6699264ca845b9c \\
Pure Prop Address: & TMFfTTpCGWMgKptEhmq4H5u6A1pxWeCHZWr \\
Theory Prop Id: & ed53da6a8a942b087f817d785b79f257de9cb0f526d77897b831913b7bfb6d4f \\
Theory Prop Address: & TMdLwNGovgqDXAVmQmmx97gnpUHqbCcATFy
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoEq_rel_strict_lowerbd
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfchfgmgpfdhehchjgdgehpfmgpghhfgchcgeg} 
$$
\begin{array}{c}
{\unicdcdadad R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{p}}}}~{{q}}}}}{\unicdbdjdcd}\\
{{{{{{{{{\mname{{PNo\_rel\_strict\_lowerbd}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{\mname{{PNo\_rel\_strict\_lowerbd}}}}~{{R}}}}~{{\alpha}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 76080ee1804cd6f07353c102ecc22dbe6f88819fba1748db34678b3a3205ce51 \\
Pure Prop Address: & TMRaaoK9aPyvyCQEyS6u7CuxJeMRarcqdua \\
Theory Prop Id: & 614e589383386bf6bffd56988d8ed052f11fa7717c9bfc928aa218b49ba14436 \\
Theory Prop Address: & TMN9uHHDyxEb7wZLV54NYgKUBnfgd8Syeiz
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_rel_strict_lowerbd_antimon
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfchfgmgpfdhehchjgdgehpfmgpghhfgchcgegpfbgogehjgngpgog} 
$$
\begin{array}{c}
{\unicdcdadad R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{\unicdcdadad \beta \in }{{{\alpha}}}{ .  }\\
{{{{{{{{{\mname{{PNo\_rel\_strict\_lowerbd}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{\mname{{PNo\_rel\_strict\_lowerbd}}}}~{{R}}}}~{{\beta}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 27f253b3d87846cda9c59a20ef82f3175140815d3b9625c2b5dfe1a332086817 \\
Pure Prop Address: & TMQ6kfs1zMTGEcR3v2oD9G2cmxvPeeGcygk \\
Theory Prop Id: & 344b63ede082390378753476ed8b39a6cdcbc2522581fb25d8b3e2dbcd366fd5 \\
Theory Prop Address: & TMVV2kTQbx2zWgUFoViphDG4PSygzxibjvw
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoEq_rel_strict_imv
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfchfgmgpfdhehchjgdgehpfjgnggh} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{p}}}}~{{q}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e3f251299cd1135da075622be47bde557bbba6cbac1e28d78bb914fdda99a216 \\
Pure Prop Address: & TMX9qa29bL95NMwjxy6sv4Ubp6sbR8ZXSAy \\
Theory Prop Id: & 7ae13df8bdc5ae35409ee041f3bcf55387bae17d0117e5e6491d3cb5cb89d4f9 \\
Theory Prop Address: & TMWRDDHJba821Tbd4gG36AUGcRoP4wgdEtZ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_rel_strict_imv_antimon
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfchfgmgpfdhehchjgdgehpfjgngghpfbgogehjgngpgog} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{\unicdcdadad \beta \in }{{{\alpha}}}{ .  }\\
{{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\beta}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2434dc0ac445139024cddcec7dd72380f70b6c60cbf7e3eea162062ef8ef30b5 \\
Pure Prop Address: & TMTzSVmWpr6G9xpGB9xRz9GHutphxoFHpm5 \\
Theory Prop Id: & 87e0ce7b41f81ec7cd5cb539c8d29a4ee84162503f0111c69851aa5cb6250f67 \\
Theory Prop Address: & TMdHPiqFbrqqnJZvUEgGaaNprnFuirYL2XJ
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgpfchfgmgpfdhehchjgdgehpffhogjgbhpfjgnggh}
  We define $\mname{PNo\_rel\_strict\_uniq\_imv}$ to be
  $$
  \begin{array}{c}
    \uniddcgcg L R \alpha p .  {{{{{\mname{PNo\_rel\_strict\_imv}}~{L}}~{R}}~{\alpha}}~{p}}\unicdcdcdhd\\
               {\unicdcdadad q : {{\iota}\unicdbdjdcd{o}} .  {{{{{{\mname{PNo\_rel\_strict\_imv}}~{L}}~{R}}~{\alpha}}~{q}}\unicdbdjdcd{{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{q}}}}
  \end{array}
  $$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 23c9d918742fceee39b9ee6e272d5bdd5f6dd40c5c6f75acb478f608b6795625 \\
Pure Object Address: & TMcVvujMz8D899TFXW92xtjqAjrJ7xZ5wns \\
Theory Object Id: & 56ed1b924504c7b2800fa12637e1e0a522196c5a1a82a00af10c643a8a5e9e6f \\
Theory Object Address: & TMJg15q9kSMojiP1K5gjvfgbDB2SFT15THF
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afoepgpfchfgmgpfdhehchjgdgehpfdhahmgjgehpfjgnggh}
  We define $\mname{PNo\_rel\_strict\_split\_imv}$ to be
  $$
  \begin{array}{c}
    \uniddcgcg L R \alpha p .  {{{{{\mname{PNo\_rel\_strict\_imv}}~{L}}~{R}}~{({{\mname{ordsucc}}~{\alpha}})}}~{({\uniddcgcg \delta .  {{{p}~{\delta}}\unicdcdcdhd{{\delta}\unicdcdgdad{\alpha}}}})}}\unicdcdcdhd\\
               {{{{{\mname{PNo\_rel\_strict\_imv}}~{L}}~{R}}~{({{\mname{ordsucc}}~{\alpha}})}}~{({\uniddcgcg \delta .  {{{p}~{\delta}}\unicdcdcdid{{\delta}={\alpha}}}})}}
  \end{array}
  $$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 217d179d323c5488315ded19cb0c9352416c9344bfdfbb19613a3ee29afb980d \\
Pure Object Address: & TMcNiZGAEhYLnFaHRVtxA3RMJ6rQRD68crC \\
Theory Object Id: & dd9ceee1f10e584685cd8a5d181d107a00ffeb879bddd77bb943edc9d869a262 \\
Theory Object Address: & TMaSwGSK92j2THPci27ww2urWPVb7Y8PaRX
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNo_extend0_eq
\end{verbatim}
\begin{theorem}\label{thm:afoepgpffgihehfgogegadpffgbh} ${\unicdcdadad \alpha .  {\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{({\uniddcgcg \delta .  {{{p}~{\delta}}\unicdcdcdhd{{\delta}\unicdcdgdad{\alpha}}}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2d441ed2f6faaa3204d9c074c159ce10040fd6680f3c4f69726e8dbc60efc1b8 \\
Pure Prop Address: & TMWBZZ2LzEnHQnhQ2EUjYaKDpHL6urz6cT8 \\
Theory Prop Id: & 5ed08e400c3976db41fd77c3d882770ae3d2bc85093f84c0060820655b4328c3 \\
Theory Prop Address: & TMPWoqhrnorzigPXmr4GmgWiGibdw73sjz8
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_extend1_eq
\end{verbatim}
\begin{theorem}\label{thm:afoepgpffgihehfgogegbdpffgbh} ${\unicdcdadad \alpha .  {\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{{{\mname{PNoEq\_}}~{\alpha}}~{p}}~{({\uniddcgcg \delta .  {{{p}~{\delta}}\unicdcdcdid{{\delta}={\alpha}}}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8eb99c36aa09b197a65897ed80cdf2570eba4e719d286f36fc4bc09afd4f3689 \\
Pure Prop Address: & TMQEFMAonmrWaLGhniLUme237vwrY6AwX3L \\
Theory Prop Id: & 4cac1a7bb678742b042b74050de5270cd3a104cca4e451080d85e20b33bff7dc \\
Theory Prop Address: & TMGkMWYd35nspxAzshHYLLW93mSrEQa177W
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_rel_imv_ex
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfchfgmgpfjgngghpffgih} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{({{\unicdcdaddd p : {{{{\iota}}\unicdbdjdcd{{o}}}} .  {{{{{{{{{{\mname{{PNo\_rel\_strict\_uniq\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}})}}\\
    \unicdcdcdid{{({{\unicdcdaddd \tau \in {{\alpha}} .  {{\unicdcdaddd p : {{{{\iota}}\unicdbdjdcd{{o}}}} .  {{{{{{{{{{\mname{{PNo\_rel\_strict\_split\_imv}}}}~{{L}}}}~{{R}}}}~{{tau}}}}~{{p}}}}}}}})}}.\\

\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3e3224100da33bad94d660a0cfa8843fa7519fd5ff75e4c23da58e5b68707714 \\
Pure Prop Address: & TMb7LjGHosQp5Zu2ZyhehFbFtpyZL2yxrkB \\
Theory Prop Id: & 749b63b90f39e599387d66934130ae6ac3e08e4f9989ca927eec36e855e09fc8 \\
Theory Prop Address: & TMMKoMsquCSh1xogZaGrGcBP6CoCXQLGM1t
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgpfmgfgogcgegeg}
  We define $\mname{PNo\_lenbdd}$ to be
  $${\uniddcgcg \alpha L .  {\unicdcdadad \beta .  {\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{{{L}~{\beta}}~{p}}\unicdbdjdcd{{\beta} \in {\alpha}}}}}}$$
  of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 009fa745aa118f0d9b24ab4efbc4dc3a4560082e0d975fd74526e3c4d6d1511f \\
Pure Object Address: & TMYM1fGNqp8awnq5MVCDqBVSdy2y4KSQX4c \\
Theory Object Id: & a452a633e2734cc4531287f806932a19db788c9ebf515fd5c65361a126fcccda \\
Theory Object Address: & TMc7MFWWHbWct8gGwr9Jjykxysxajx7WxYR
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNo_lenbdd_strict_imv_extend0
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfmgfgogcgegegpfdhehchjgdgehpfjgngghpffgihehfgogegad} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{({{{{\mname{{ordsucc}}}}~{{\alpha}}}})}}}}~{{({{\uniddcgcg \delta .  {{{{{{p}}~{{\delta}}}}\unicdcdcdhd{{{{\delta}}\unicdcdgdad{{\alpha}}}}}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 17a11416cb52c3c428daf77fd88095938feda8129ca2851349863fa5a7f70fab \\
Pure Prop Address: & TMZ7dVvvzt9aBgJceuie2x9CTj7zzynBVKW \\
Theory Prop Id: & 69327e8aa4dfac05b5218d04f861172313490347f560cc3142dc11e85c3d1de9 \\
Theory Prop Address: & TMFuqsPkUmPBmPBLFudNnuZaej9uSQJFdHA
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_lenbdd_strict_imv_extend1
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfmgfgogcgegegpfdhehchjgdgehpfjgngghpffgihehfgogegbd} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{({{{{\mname{{ordsucc}}}}~{{\alpha}}}})}}}}~{{({{\uniddcgcg \delta .  {{{{{{p}}~{{\delta}}}}\unicdcdcdid{{{{\delta}}={{\alpha}}}}}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ad69ec2daa15891f8d7b060e461a3f1b3e63589a17d9fe0b50275b428c54760d \\
Pure Prop Address: & TMHJrXcf4TaXkTgTPiRVvo9nkWreomZdRJn \\
Theory Prop Id: & 5ff9936fa7da19e574c1e407d5dc4601e10f51aca435de66bb11cc9e7ada1df8 \\
Theory Prop Address: & TMPvUJKpgzdwMRuCFNU2GRtAfonYYsF38vv
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_lenbdd_strict_imv_split
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfmgfgogcgegegpfdhehchjgdgehpfjgngghpfdhahmgjgeh} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNo\_rel\_strict\_split\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 76ae4f9f794260b3f0f83c999d5560105286744cef38b1a0234287b13ad8db9e \\
Pure Prop Address: & TMaW9z99uWRswWnxdDU9cvmKc6HtRaCtVRU \\
Theory Prop Id: & f98998c738859a6ff5fbcf1701155ca36cbcb0caef56b13a9cff1b1c937b6ece \\
Theory Prop Address: & TMFZtcp8Jrq2c5Vbiocbri3d5UhPuowpAaw
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_rel_imv_bdd_ex
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfchfgmgpfjgngghpfcgegegpffgih} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}\\
{{\unicdcdaddd \beta \in {{{{\mname{{ordsucc}}}}~{{\alpha}}}} .  {{\unicdcdaddd p : {{{{\iota}}\unicdbdjdcd{{o}}}} .  {{{{{{{{{{\mname{{PNo\_rel\_strict\_split\_imv}}}}~{{L}}}}~{{R}}}}~{{\beta}}}}~{{p}}}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9a98e4a421cb8eedc40a867b16cac1d2529c4dfabcc3ff3b451f9e220420f209 \\
Pure Prop Address: & TMRSz1Na7BW752MS6vDye8Y6Vo8SfeVhV1j \\
Theory Prop Id: & 25bcd2bc2f9d34e476f5e18c7b51c9075ed93e802655cc620f888c8e1fd16a22 \\
Theory Prop Address: & TMQfEm1NLf1eHjLsYL7HyTLrAxiyrsEBsSQ
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgpfdhehchjgdgehpffhahahfgchcgeg}
  We define $\mname{PNo\_strict\_upperbd}$ to be
  $${\uniddcgcg L \alpha p .  {\unicdcdadad \beta .  {{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{\unicdcdadad q : {{\iota}\unicdbdjdcd{o}} .  {{{{L}~{\beta}}~{q}}\unicdbdjdcd{{{{{\mname{PNoLt}}~{\beta}}~{q}}~{\alpha}}~{p}}}}}}}$$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 6913bd802b6ead043221cf6994931f83418bb29a379dc6f65a313e7daa830dcc \\
Pure Object Address: & TMKnMYekteuHoeWstJHEmJ4JAdKPexnFA42 \\
Theory Object Id: & 3723abda991ef19534694f8f4cd239c37f64fc7c8d2f1cbae883cc9f9d0d69b3 \\
Theory Object Address: & TMRkoJnHGKhQmpq21wXvTR2RBkLqLSagigX
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afoepgpfdhehchjgdgehpfmgpghhfgchcgeg}
  We define $\mname{PNo\_strict\_lowerbd}$ to be
  $${\uniddcgcg R \alpha p .  {\unicdcdadad \beta .  {{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{\unicdcdadad q : {{\iota}\unicdbdjdcd{o}} .  {{{{R}~{\beta}}~{q}}\unicdbdjdcd{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}}}}}}$$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 87fac4547027c0dfeffbe22b750fcc4ce9638e0ae321aeaa99144ce5a5a6de3e \\
Pure Object Address: & TMZjYeWB7UQog19EjFL4PmaNhnokNCuMGJV \\
Theory Object Id: & 3606d45e76891d3b7e28745aa1b61986d255603f7abeb8f63a847b1ccc62b95d \\
Theory Object Address: & TMXvc4Vsa77sgNvdnXny3CP9WCzsnoarmV1
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afoepgpfdhehchjgdgehpfjgnggh}
We define $\mname{PNo\_strict\_imv}$ to be $${\uniddcgcg L R \alpha p .  {{{{{\mname{PNo\_strict\_upperbd}}~{L}}~{\alpha}}~{p}}\unicdcdcdhd{{{{\mname{PNo\_strict\_lowerbd}}~{R}}~{\alpha}}~{p}}}}$$ of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & ee97f6fc35369c0aa0ddf247ea2ee871ae4efd862b662820df5edcc2283ba7e3 \\
Pure Object Address: & TMbEV7hkJGaPtHGWT7FsCyrBxQjHqPm9Ct9 \\
Theory Object Id: & bc6c55650a1ceb99aefb5c3b39fed27e70fc45b811cc09f91f1fd23a0cfbe326 \\
Theory Object Address: & TMHAeW8tAPaNuMKzRy2t7UZyujPe3oz8chT
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoEq_strict_upperbd
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfdhehchjgdgehpffhahahfgchcgeg} 
$$
\begin{array}{c}
{\unicdcdadad L : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{p}}}}~{{q}}}}}{\unicdbdjdcd}\\
{{{{{{{{{\mname{{PNo\_strict\_upperbd}}}}~{{L}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{PNo\_strict\_upperbd}}}}~{{L}}}}~{{\alpha}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9b7b8e2913af3ee84872e0fd4fca5bfeb65de5ae332f50e8005ce0b40c0390d1 \\
Pure Prop Address: & TMMwWqqBxTemQcZaoYuQ5csnpPswRJuhRvB \\
Theory Prop Id: & f28b04874428203d54a6576e08aded8841fea8787b265c71cffb0a98a190b2b1 \\
Theory Prop Address: & TMFSxkkKeWx8CjszfDWMQmeEXEdqcpXHZqL
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoEq_strict_lowerbd
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfdhehchjgdgehpfmgpghhfgchcgeg} 
$$
\begin{array}{c}
{\unicdcdadad R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{p}}}}~{{q}}}}}{\unicdbdjdcd}\\
{{{{{{{{{\mname{{PNo\_strict\_lowerbd}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{PNo\_strict\_lowerbd}}}}~{{R}}}}~{{\alpha}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4db3f9364356674211f51fc9a183e838dff60a313c518ca7d65b13fd0698d5a1 \\
Pure Prop Address: & TMG1CzmnW8AfKeL7zGEwMEtn4b1ApJy6q1g \\
Theory Prop Id: & 67fa73a4227fa42a9ed6db8fd115f876749a15c619bbceedaf8114345d2a228b \\
Theory Prop Address: & TMFvqJhbAHTTaVpfbgHnnHh9Roc9bwF9YcE
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoEq_strict_imv
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfdhehchjgdgehpfjgnggh} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{p}}}}~{{q}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}{{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1ec3525cfde857873a05b1492e442895aa98e91b1a7a58db58116a8baa7863f3 \\
Pure Prop Address: & TMWLyiXz4WsN9qiqmYUTwRb7RrR4sGpvULn \\
Theory Prop Id: & 5f582f95ff13676a0730e043b393f09bf9b302f7c01959e268fdbaeaaba7bd23 \\
Theory Prop Address: & TMZZL19r37MLXbo9tRES9F3kF1Cdd7rP1Vx
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_strict_upperbd_imp_rel_strict_upperbd
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfdhehchjgdgehpffhahahfgchcgegpfjgngahpfchfgmgpfdhehchjgdgehpffhahahfgchcgeg} 
$$
\begin{array}{c}
{\unicdcdadad L : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad \beta \in }{{{{{\mname{{ordsucc}}}}~{{\alpha}}}}}{ .  }{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{{{{{{{\mname{{PNo\_strict\_upperbd}}}}~{{L}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{PNo\_rel\_strict\_upperbd}}}}~{{L}}}}~{{\beta}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2898f0821b6ee4c20a8875fb97191a1b12a50a1e963c11f9aa1a31c2bd46f8e2 \\
Pure Prop Address: & TMXJEXJKF7JrSAhmPixqwmaahA97oWAoeLL \\
Theory Prop Id: & 3cdd3b4f20ab67c1be5329318323deafeea009f34f28baa01090782c3bdfdd0e \\
Theory Prop Address: & TMbeJVU3GyejPAEuuCqdu9srTvCD2PVUAy8
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_strict_lowerbd_imp_rel_strict_lowerbd
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfdhehchjgdgehpfmgpghhfgchcgegpfjgngahpfchfgmgpfdhehchjgdgehpfmgpghhfgchcgeg} 
$$
\begin{array}{c}
{\unicdcdadad R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad \beta \in }{{{{{\mname{{ordsucc}}}}~{{\alpha}}}}}{ .  }{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{{{{{{{\mname{{PNo\_strict\_lowerbd}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{PNo\_rel\_strict\_lowerbd}}}}~{{R}}}}~{{\beta}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 442af59a4f76dc1ad25be2cc8abd5af2847bff7cd5f1b08ec480d479fe6f9148 \\
Pure Prop Address: & TMMPUoKVYbXX3DBLQHFe5h9b2gv8eiAFaAb \\
Theory Prop Id: & 04acc2c7664cb2baa2edfe5f073538b0cb855556d54d07337c5d7f3ccfcf5ab0 \\
Theory Prop Address: & TMPPvnYPwEv5KBauhbm93s9dRpjMraJ2RMs
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_strict_imv_imp_rel_strict_imv
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfdhehchjgdgehpfjgngghpfjgngahpfchfgmgpfdhehchjgdgehpfjgnggh} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad \beta \in }{{{{{\mname{{ordsucc}}}}~{{\alpha}}}}}{ .  }{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}{{{{{{{{{{\mname{{PNo\_rel\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\beta}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f4fc71c90b443abe5f08ec2e1cbad2b8e178220ae7a28146d2debaa27c8dca9e \\
Pure Prop Address: & TMUZW37y1v2yvXxB4qDRF8ycQ12jacn99PZ \\
Theory Prop Id: & 46a6aacddbb0224f249308bdbd8f4155dd2574ab91755a3faeebd246dd1ac8da \\
Theory Prop Address: & TMaWoAQeNZDif3qPS3T1yTknjm8pRRW8a4h
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_rel_split_imv_imp_strict_imv
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfchfgmgpfdhahmgjgehpfjgngghpfjgngahpfdhehchjgdgehpfjgnggh} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{{{{{{{{{\mname{{PNo\_rel\_strict\_split\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 72be36124095b1c454cfc103ace8e578f6da1f3537bed3a63dff48a5ee9e67e3 \\
Pure Prop Address: & TMGERrdQ6q2bYZah7DP6dYkxMqTrBkNs3qx \\
Theory Prop Id: & adfdb01bca46911cd3ac3a75bf022e1e4e0316883433f9218fddedd9a12aceeb \\
Theory Prop Address: & TMTEYb8yRqBikXo6Wt5MPUkSefJ4iSEG83i
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_PNo_strict_imv
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfafoepgpfdhehchjgdgehpfjgnggh} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{({{\unicdcdadad \beta \in {{\alpha}} .  {{{{p}}~{{\beta}}}}}})}}}{\unicdbdjdcd}\\
{{{({{\unicdcdadad \beta .  {{{{{{\mname{{ordinal}}}}~{{\beta}}}}\unicdbdjdcd{{\unicdcdadad q : {{{{\iota}}\unicdbdjdcd{{o}}}} .  {{{{{{{{L}}~{{\beta}}}}~{{q}}}}\unicdbdjdcd{{{{\beta}} \in {{\alpha}}}}}}}}}}}})}}}{\unicdbdjdcd}{{{({{\unicdcdadad \beta \in {{\alpha}} .  {{{{{{L}}~{{\beta}}}}~{{p}}}}}})}}}{\unicdbdjdcd}\\
{{{({{\unicdcdadad \beta .  {{{{{{\mname{{ordinal}}}}~{{\beta}}}}\unicdbdjdcd{{\unicdcdadad q : {{{{\iota}}\unicdbdjdcd{{o}}}} .  {{\preoh{{{{{{R}}~{{\beta}}}}~{{q}}}}}}}}}}}})}}}{\unicdbdjdcd}{{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6ac14c06740c6ba05ec4cd77fc370b4915f44952ab2e3e843377f3ae0f4ef372 \\
Pure Prop Address: & TMPF3ri47F9YTagy21jCuzhVZiWgwJhBagV \\
Theory Prop Id: & d6a73528e480611cb4cc96b7f8a095233cd0b5ce7a4c96d973c70a42ac001c03 \\
Theory Prop Address: & TMTeL2CwHrU7yAsCK9SoepcoGiqPerjGfAP
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_lenbdd_strict_imv_ex
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfmgfgogcgegegpfdhehchjgdgehpfjgngghpffgih} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}\\
{{\unicdcdaddd \beta \in {{{{\mname{{ordsucc}}}}~{{\alpha}}}} .  {{\unicdcdaddd p : {{{{\iota}}\unicdbdjdcd{{o}}}} .  {{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\beta}}}}~{{p}}}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9485d658371746bd448a504cec578379db903809a34d0f3d7395e9ad92f52bc5 \\
Pure Prop Address: & TMboc8u3wjuBmQSkALzv47nZjKdu9Ln6G1i \\
Theory Prop Id: & a864a57867c4b8fa4046e9c687fd4f648d42ccbee4cef69cf3b1c28bb983c0eb \\
Theory Prop Address: & TMXJMwoh8T8TYKwFJYwWGnhaeosMEU67QtG
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgpfmgfgbgdhehpfchfgah}
  We define $\mname{PNo\_least\_rep}$ to be
  $$
  \begin{array}{c}
    \uniddcgcg L R \beta p .  {{{\mname{ordinal}}~{\beta}}\unicdcdcdhd{{{{{\mname{PNo\_strict\_imv}}~{L}}~{R}}~{\beta}}~{p}}}\unicdcdcdhd\\
      {\unicdcdadad \gamma \in {\beta} .  {\unicdcdadad q : {{\iota}\unicdbdjdcd{o}} .  {\preoh{{{{{{\mname{PNo\_strict\_imv}}~{L}}~{R}}~{\gamma}}~{q}}}}}}
  \end{array}
  $$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 0711e2a692a3c2bdc384654882d1195242a019a3fc5d1a73dba563c4291fc08b \\
Pure Object Address: & TMMrUbZHNUQojuJFRTNTGugMnkz3yaWyJjG \\
Theory Object Id: & e4c707e63a9d11d1b61d0db26f0cf3155ea6db0eca2d2f3dbf812d36a70e6c70 \\
Theory Object Address: & TMdTiKpkh6t2DH3NuLyGfeGX79J5YUPNSBs
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNo_lenbdd_least_rep_ex
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfmgfgogcgegegpfmgfgbgdhehpfchfgahpffgih} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}\\
{{\unicdcdaddd \beta .  {{\unicdcdaddd p : {{{{\iota}}\unicdbdjdcd{{o}}}} .  {{{{{{{{{{\mname{{PNo\_least\_rep}}}}~{{L}}}}~{{R}}}}~{{\beta}}}}~{{p}}}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4541c94e58f35924a0da5fd80f4d2327c22d662287cf6c55c9962fcacd32b6c0 \\
Pure Prop Address: & TMV8v9NRJQArdep3t94cASip2uDW46XdwHJ \\
Theory Prop Id: & fc95a5ba7826e45c22086496d12170955e43c43c454f7c838a38598d83edd933 \\
Theory Prop Address: & TMQu9hsS7YcjLuJMR9UBzRz4gDzHhbZw56N
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgpfmgfgbgdhehpfchfgahcd}
  We define $\mname{PNo\_least\_rep2}$ to be
  $${\uniddcgcg L R \beta p .  {{{{{{\mname{PNo\_least\_rep}}~{L}}~{R}}~{\beta}}~{p}}\unicdcdcdhd{\unicdcdadad x .  {{{x}\unicdcdadjd{\beta}}\unicdbdjdcd{\preoh{{{p}~{x}}}}}}}}$$
  of type ${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 1f6dc5b1a612afebee904f9acd1772cd31ab0f61111d3f74cfbfb91c82cfc729 \\
Pure Object Address: & TMTPNGLhRLywGV7b2VErKKRqsBtTRqSdE63 \\
Theory Object Id: & 09d320f50b226d6de787cb7f04c0ac3eedf43d3981566d46692b73e884662b9b \\
Theory Object Address: & TMRAk8jd2vN252xuwfLzStCvRTrSjBsrXz3
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNo_strict_imv_pred_eq
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfdhehchjgdgehpfjgngghpfahchfgegpffgbh} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{{{{{{{{{\mname{{PNo\_least\_rep}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{p}}}}}{\unicdbdjdcd}{{{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\alpha}}}}~{{q}}}}}{\unicdbdjdcd}{\unicdcdadad \beta \in }{{{\alpha}}}{ .  }\\
{{{{{{p}}~{{\beta}}}}\unicdbdjded{{{{q}}~{{\beta}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5e7822daeca655e2242d474ec86dd9579effe366031b4f9b73ec5ca4a659a545 \\
Pure Prop Address: & TMZq9y1Uq2aENtjsxC2H2kmR5Sts8dSVeAn \\
Theory Prop Id: & 72e14d37cd81d2a456f9878f785f26768d709f7848a4cf627865ba0c3662b1b3 \\
Theory Prop Address: & TMP8YWfHwKekxFLMQmDee1eb36HPVWfDCkn
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_lenbdd_least_rep2_exuniq2
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfmgfgogcgegegpfmgfgbgdhehpfchfgahcdpffgihfhogjgbhcd} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}\\
\unicdcdaddd \beta .  {{({{\unicdcdaddd p : {{{{\iota}}\unicdbdjdcd{{o}}}} .  {{{{{{{{{{\mname{{PNo\_least\_rep2}}}}~{{L}}}}~{{R}}}}~{{\beta}}}}~{{p}}}}}})}}\unicdcdcdhd\\
(\unicdcdadad p q : {{{{\iota}}\unicdbdjdcd{{o}}}} .  {{{{{{{{{{\mname{{PNo\_least\_rep2}}}}~{{L}}}}~{{R}}}}~{{\beta}}}}~{{p}}}}\unicdbdjdcd\\
    {{{{{{{{{{{{\mname{{PNo\_least\_rep2}}}}~{{L}}}}~{{R}}}}~{{\beta}}}}~{{q}}}}\unicdbdjdcd{{{{p}}={{q}}}}}}).\\

\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5060a91293273a3d3416ab99070c6d1e15f0cbef0c24aa15399ee37c868f691c \\
Pure Prop Address: & TMT7sUcy4ndNzr9t94F3xrCG638UAh62av3 \\
Theory Prop Id: & 8b62c56d475417d4d361eaf3c3c3680d80138732014122d77d426f4860f69081 \\
Theory Prop Address: & TMNmdxAHFQ1AVyvBStP6JJjmC4WocYjvTV2
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgpfcgeg}
$\mname{PNo\_bd}$ is the opaque object of type $${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{\iota}}}$$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & ed76e76de9b58e621daa601cca73b4159a437ba0e73114924cb92ec8044f2aa2 \\
Pure Object Address: & TMNBrUd5KSYmRWHN49oFdMFsUPTiSKvsa9Q \\
Theory Object Id: & aa75efbc66bb94911a3c3b156ea65bbe1775870ce7a06e24c056f8d5cce3660d \\
Theory Object Address: & TMV6kaag6YpB5pyMDkL4wftUcq2Diskvi7a
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afoepgpfahchfgeg}
$\mname{PNo\_pred}$ is the opaque object of type $${{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}}$$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & b2d51dcfccb9527e9551b0d0c47d891c9031a1d4ee87bba5a9ae5215025d107a \\
Pure Object Address: & TMU5awsv39apDkXc2r3HzJRP37vuRruKsnf \\
Theory Object Id: & 2efd18fb25499ea37fc2af1b7a0ffdd36e6abc401c42458854e4f4288f7ca530 \\
Theory Object Address: & TMStxoqH3zLresqVCQdXwQ55q36aVygMDeL
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNo_bd_pred_lem
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfcgegpfahchfgegpfmgfgng} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNo\_least\_rep2}}}}~{{L}}}}~{{R}}}}~{{({{{{{{\mname{{PNo\_bd}}}}~{{L}}}}~{{R}}}})}}}}~{{({{{{{{\mname{{PNo\_pred}}}}~{{L}}}}~{{R}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 749b8f4e860ae9a289350b300230645c0611ffff60798a2d0cdfd7fe0476bd2a \\
Pure Prop Address: & TMYJBHCdV3Ct6gBy3MMb8STjafZb3SviQ9v \\
Theory Prop Id: & 5de4a33591f164add23849f5545d9322fa8885bd09e7ba1894ce774b2c9cab1a \\
Theory Prop Address: & TMP6BAwWzoNKb47BSVYq4GJmL3WoFMQGbqf
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_bd_pred
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfcgegpfahchfgeg} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNo\_least\_rep}}}}~{{L}}}}~{{R}}}}~{{({{{{{{\mname{{PNo\_bd}}}}~{{L}}}}~{{R}}}})}}}}~{{({{{{{{\mname{{PNo\_pred}}}}~{{L}}}}~{{R}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 42224b3374ae5d0ed32e9420c84f18f8ea84d9eb8daae7337d3905a179b65d65 \\
Pure Prop Address: & TMR9wZusZqBwrBJNUbqyESMWattt5sL41Jf \\
Theory Prop Id: & 3309f6a3ddcb2319d01432fd7a6a7d3c05f3b9836222f520fb3ec000b2b7299f \\
Theory Prop Address: & TMH4SyZ8pRU9WGJb2eb6yLYAQnwpF555Lo5
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_bd_ord
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfcgegpfpgcheg} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}{{{{\mname{{ordinal}}}}~{{({{{{{{\mname{{PNo\_bd}}}}~{{L}}}}~{{R}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b40a91e75a7dcc94d55b6763376f5f366ab4a140e5e8b0380656e21f2b419fb8 \\
Pure Prop Address: & TMZxHtMby9p5j448ESrb1iGeSp3QjaX8APB \\
Theory Prop Id: & bc6e649a31db5eaff908ac0cc857d64018672dfb788fc1bfef50fbd2a45f597c \\
Theory Prop Address: & TMYrVH5VdsuG5TzNCVy7afWTZ3eHsgNAjH3
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_bd_pred_strict_imv
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfcgegpfahchfgegpfdhehchjgdgehpfjgnggh} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{({{{{{{\mname{{PNo\_bd}}}}~{{L}}}}~{{R}}}})}}}}~{{({{{{{{\mname{{PNo\_pred}}}}~{{L}}}}~{{R}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b6398cfc0d268e1265cb766e4a3c763db5b5aba648cb2602837dfed49a6a1373 \\
Pure Prop Address: & TMVx2aGZb6qmcJNfMJMoPgY2mzk5RmNZstN \\
Theory Prop Id: & ee0997fab4c7c13a66f2eda4601d185456ed98ab1a2c0269f8b7a47879f0a300 \\
Theory Prop Address: & TMMZTkgpVapPLqBQTVn8NSVkQRWNow8gRiS
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_bd_least_imv
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfcgegpfmgfgbgdhehpfjgnggh} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \gamma \in }{{{{{{{\mname{{PNo\_bd}}}}~{{L}}}}~{{R}}}}}{ .  }{\unicdcdadad q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{\preoh{{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{L}}}}~{{R}}}}~{{\gamma}}}}~{{q}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8756915afa703c80eca22a6825041ae9983fe1efe0425560dd4d0e65970d8a91 \\
Pure Prop Address: & TMEsGe3wG8yXWS94JJYUXQFxvwvV78FGr2C \\
Theory Prop Id: & 266277c31581e02700387ec68a784741f97e44197639a61abe33b4bce3c8eccb \\
Theory Prop Address: & TMR3RGTyTYH8yzKLkmxhSTiEdgD4ZTKaapR
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNo_bd_In
\end{verbatim}
\begin{theorem}\label{thm:afoepgpfcgegpfjeog} 
$$
\begin{array}{c}
{\unicdcdadad L R : }{{{{{\iota}}\unicdbdjdcd{{{{({{{{\iota}}\unicdbdjdcd{{o}}}})}}\unicdbdjdcd{{o}}}}}}}{ .  }{{{{{{{\mname{{PNoLt\_pwise}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}\\
{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{L}}}}}{\unicdbdjdcd}{{{{{{{\mname{{PNo\_lenbdd}}}}~{{\alpha}}}}~{{R}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{PNo\_bd}}}}~{{L}}}}~{{R}}}} \in {{{{\mname{{ordsucc}}}}~{{\alpha}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ad857b445d1196a6cfc19a14ac9d45dff7f3b25ed55d9764dcbb9d4d5efdea8d \\
Pure Prop Address: & TMasNPNdz1vv1jt29FCFyQpkupqkrcwKcUy \\
Theory Prop Id: & 4719e12b28711d9d989a9ed9afa168f501d090c55ff5f971c2a9c4b951a3a4c8 \\
Theory Prop Address: & TMZ1d5WRxZPZR9eT7BLzoFprMn6mNkfxFcZ
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:afoepgdefhehme}
We define $\mname{PNoCutL}$ to be ${\uniddcgcg \alpha p \beta q .  {{{\beta} \in {\alpha}}\unicdcdcdhd{{{{{\mname{PNoLt}}~{\beta}}~{q}}~{\alpha}}~{p}}}}$ of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & d272c756dc639d0c36828bd42543cc2a0d93082c0cbbe41ca2064f1cff38c130 \\
Pure Object Address: & TMW89BeZ4LwBAWUSvcHTTffq4NjA9eURfe5 \\
Theory Object Id: & 5c2353f1a9132e4b0bef01f9d3b48e3dd377089926d3ec56a9c1b5f62d18564e \\
Theory Object Address: & TMXddSCNaHGduK9qNYrg5YeV7BsECCYxzJL
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afoepgdefhehcf}
We define $\mname{PNoCutR}$ to be ${\uniddcgcg \alpha p \beta q .  {{{\beta} \in {\alpha}}\unicdcdcdhd{{{{{\mname{PNoLt}}~{\alpha}}~{p}}~{\beta}}~{q}}}}$ of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{o}}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & ff28d9a140c8b282282ead76a2a40763c4c4800ebff15a4502f32af6282a3d2e \\
Pure Object Address: & TMQ43uDGqp316dHj1xW6txbDkdt4BerVFRg \\
Theory Object Id: & 6be9d2eca9c688e6504e0251a1344ee158dafeb178c09a30d02481f120ec9fdd \\
Theory Object Address: & TMJvHmHmkfm3PZnD7HiJxv464ZKQNcn3xK4
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoCutL_lenbdd
\end{verbatim}
\begin{theorem}\label{thm:afoepgdefhehmepfmgfgogcgegeg} ${\unicdcdadad \alpha .  {\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{{\mname{PNo\_lenbdd}}~{\alpha}}~{({{{\mname{PNoCutL}}~{\alpha}}~{p}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2ce3cac589e0288fcb056a351d7b7f52ba86fb73ca736e136679ff9ef0ea1ac3 \\
Pure Prop Address: & TMVhtYXkotNZng4p4VbWeUpqAzgiFcEJAs2 \\
Theory Prop Id: & 8d7ea3b990fcb5a72b452b48eff84626931337892eb12d3a2f14176e6d33964d \\
Theory Prop Address: & TMbipKXeU4Z2DY2jqE2CDGdVBX2xtErK7wW
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoCutR_lenbdd
\end{verbatim}
\begin{theorem}\label{thm:afoepgdefhehcfpfmgfgogcgegeg} ${\unicdcdadad \alpha .  {\unicdcdadad p : {{\iota}\unicdbdjdcd{o}} .  {{{\mname{PNo\_lenbdd}}~{\alpha}}~{({{{\mname{PNoCutR}}~{\alpha}}~{p}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & abe446408190f6d442a1f2eab7727c53458665db08e199262714d89da47c7000 \\
Pure Prop Address: & TMNwdvruagyifCoqN3GpPqcvsoJxB98XesL \\
Theory Prop Id: & bba4ed960136adb5c77796704f207374e63fcf341cd9a438a50561365d6f14a0 \\
Theory Prop Address: & TMcNY1d7D2m6KMq1LqqgqprtHuNXxok7xt6
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoCut_pwise
\end{verbatim}
\begin{theorem}\label{thm:afoepgdefhehpfahhhjgdhfg} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{{{{\mname{{PNoLt\_pwise}}}}~{{({{{{{{\mname{{PNoCutL}}}}~{{\alpha}}}}~{{p}}}})}}}}~{{({{{{{{\mname{{PNoCutR}}}}~{{\alpha}}}}~{{p}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8bb0692a31e5ad0bfd5013045a2fbf8c7889e948699261b72291eae629aa15fe \\
Pure Prop Address: & TMJo1MdhfFXdWBivM8BPiej6KZWhKLTQmBM \\
Theory Prop Id: & e9740986f55e09faff6e0e851bdf5251ed23276fc5747ca8a67afcdf70970cbb \\
Theory Prop Address: & TMZSEknJujgUGg2MoFZkCMMNLvQFRFD7SYC
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoCut_strict_imv
\end{verbatim}
\begin{theorem}\label{thm:afoepgdefhehpfdhehchjgdgehpfjgnggh} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{{{{{{{{\mname{{PNo\_strict\_imv}}}}~{{({{{{{{\mname{{PNoCutL}}}}~{{\alpha}}}}~{{p}}}})}}}}~{{({{{{{{\mname{{PNoCutR}}}}~{{\alpha}}}}~{{p}}}})}}}}~{{\alpha}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & dd8b71e50df3e9a494df5a5572ce41f85b622aa24c1f3e1ff457aa4ba08718be \\
Pure Prop Address: & TMKFURgHEo6iHpwwz6JnZNJdRMvj8XUTA7D \\
Theory Prop Id: & 6e3ce431e17b20e770f06809ce5182eb248cf10acb5c38a200119fe0520c9057 \\
Theory Prop Address: & TMPNQHfdoNywBhUdkPA9hJADKARLduKe5aY
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoCut_bd_eq
\end{verbatim}
\begin{theorem}\label{thm:afoepgdefhehpfcgegpffgbh} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{{{{{{\mname{{PNo\_bd}}}}~{{({{{{{{\mname{{PNoCutL}}}}~{{\alpha}}}}~{{p}}}})}}}}~{{({{{{{{\mname{{PNoCutR}}}}~{{\alpha}}}}~{{p}}}})}}}}={{\alpha}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9dda7c4906426d138c81234f3d50f5bc4402ad268025cc5da24c9b6fb4397ad3 \\
Pure Prop Address: & TMVsjT6ktkVWYJDvH6eTEYnV2zKwFHA9xag \\
Theory Prop Id: & 8b0d28708c0a512260761ccd7dc5373cc4fbff3ff56d623c12cc98c1db4e3747 \\
Theory Prop Address: & TMR6CRmoDWEZoHBXWxwGBZgx6D5qz3GMeMa
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoCut_pred_eq
\end{verbatim}
\begin{theorem}\label{thm:afoepgdefhehpfahchfgegpffgbh} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{p}}}}~{{({{{{{{\mname{{PNo\_pred}}}}~{{({{{{{{\mname{{PNoCutL}}}}~{{\alpha}}}}~{{p}}}})}}}}~{{({{{{{{\mname{{PNoCutR}}}}~{{\alpha}}}}~{{p}}}})}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3c5499b31499c700a1fb0b68f063a4b755a2658fb92632c82d2f048d7170c0f5 \\
Pure Prop Address: & TMawNU4WyhEJede1jj4Sjss5i363KEbGi1F \\
Theory Prop Id: & 95a2d638ee2907592549cba3eecd680371133a20f157bf39a1ec7d00b4682410 \\
Theory Prop Address: & TMRmNdzKhB7GZ9dgSsWj9tM5jWyTG3QtGrZ
\end{tabular}
\end{center}

\end{theorem}
\section{Surreal Numbers as Sets} % TaggedSets
\section{TaggedSets}\label{efbghghgfgegdffgehdh}Let $tag : {{\iota}\unicdbdjdcd{\iota}}$ be ${\uniddcgcg \alpha .  {{{\mname{SetAdjoin}}~{\alpha}}~{\{{1}\}}}}$.
{\bf{Notation.}} We use $\posthc{-}$ as a postfix operator  corresponding to applying term ${tag}$.
\begin{verbatim}
not_TransSet_Sing1
\end{verbatim}
\begin{theorem}\label{thm:ogpgehpfefchbgogdhdffgehpfdfjgoghgbd} ${\preoh{{{\mname{TransSet}}~{\{{1}\}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8f2d673650ebb57c98d6d8d2a5fc915f7f2d811cbaf7858a359e1b97bb6bde17 \\
Pure Prop Address: & TMJBscGcLCjPWBHB8QVUutxyACNMVYf5z2Q \\
Theory Prop Id: & be9a70c02282395d672ab7c5e5f2551733683ad6270f097ab4c85edf46ad1a63 \\
Theory Prop Address: & TMWGSzNuxQmGiDUtn5MjVnUHh75ihtU4J6W
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
not_ordinal_Sing1
\end{verbatim}
\begin{theorem}\label{thm:ogpgehpfpgchegjgogbgmgpfdfjgoghgbd} ${\preoh{{{\mname{ordinal}}~{\{{1}\}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 99a3f6ae0ef1daee4a28fe3fcfd481e5a6573dfa220b080f34987386122d9c11 \\
Pure Prop Address: & TMUKBsrmU2hKQg3EmpkpHjq5Vz3jGSJqAGL \\
Theory Prop Id: & db797718fa26a8f2ad7c62c261b77ebfd7b9de2b25f962911447dc5838d54b9c \\
Theory Prop Address: & TMKxsifEH9Ph4ovj8YaJwts3LDF7brCPFZZ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
tagged_not_ordinal
\end{verbatim}
\begin{theorem}\label{thm:ehbghghgfgegpfogpgehpfpgchegjgogbgmg} ${\unicdcdadad y .  {\preoh{{{\mname{ordinal}}~{({\posthc{{y}}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f9488c7d361c3059b61e3d76aaa5bb7df6cb13c210dcceb3c43c7b2388c0afea \\
Pure Prop Address: & TMSUbK9326n5Ft376LsWWx4xfGaDcP52PD5 \\
Theory Prop Id: & aaf6e467102a87c8e9f73204645cc7de8b8c4dcd478bf528601bbcb36cf3a97d \\
Theory Prop Address: & TMVBkLkYkcbcmG1H2dngcHEkcuQeJt8yAkf
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
tagged_notin_ordinal
\end{verbatim}
\begin{theorem}\label{thm:ehbghghgfgegpfogpgehjgogpfpgchegjgogbgmg} ${\unicdcdadad \alpha y .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{({\posthc{{y}}})}\unicdcdadjd{\alpha}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b603ba84d8e3122663fa360bfa699dad0ab4e1f633f670e4cda297f9ba5b0208 \\
Pure Prop Address: & TMFQrda9FfP9kMHatsRA4mRhspKL6dCBanm \\
Theory Prop Id: & c0e2c350bcd1226986c6e7fdbfa0f2c19884df170a172b85b757f9de370e72a5 \\
Theory Prop Address: & TMVojt27k6SvjqAbUtQu8cJoDPMYpTdsKum
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
tagged_eqE_Subq
\end{verbatim}
\begin{theorem}\label{thm:ehbghghgfgegpffgbhfepfdffhcgbh} ${\unicdcdadad \alpha \beta .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\posthc{{\alpha}}}={\posthc{{\beta}}}}\unicdbdjdcd{{\alpha} \subseteq {\beta}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & cf032f4043866a354e9e5c9a481f6041aefa5320fd3e30ce2a8a672c5cbdc19e \\
Pure Prop Address: & TMTWKSZoPeTMh9eB7LoRwuk94Bhfw2wU5zb \\
Theory Prop Id: & ef37f58abd6ded2b344094fa0f103848b972602d46465aa33548d6c5eeffd6ed \\
Theory Prop Address: & TMdBj6QvMekfbBeqoyfr1PZdrzpAAeNC5Ek
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
tagged_eqE_eq
\end{verbatim}
\begin{theorem}\label{thm:ehbghghgfgegpffgbhfepffgbh} ${\unicdcdadad \alpha \beta .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{{\posthc{{\alpha}}}={\posthc{{\beta}}}}\unicdbdjdcd{{\alpha}={\beta}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1f7dc940e194a7544299b8233619e1f491b2dde8acde374d90cb863cd307b0b7 \\
Pure Prop Address: & TMKi9eMo4knWpH5ySxts7dqcS5g7CzxpzRQ \\
Theory Prop Id: & ec1608fa42f468b541a15ac8a164117d2d5256fb753cb09aa41258ec590a5e00 \\
Theory Prop Address: & TMYDsoSZjgQSPetekXkj4s6GU5ub3XVJaWh
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
tagged_ReplE
\end{verbatim}
\begin{theorem}\label{thm:ehbghghgfgegpfcffgahmgfe} 
$$
\begin{array}{c}
{\unicdcdadad \alpha \beta .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{\mname{{ordinal}}}}~{{\beta}}}}}{\unicdbdjdcd}{{{{{\posthc{{\beta}}}} \in {{\{{{\posthc{{\gamma}}}}|\gamma \in {{\alpha}}\}}}}}}{\unicdbdjdcd}{{{{\beta}} \in {{\alpha}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a1d43c66e0e7a187f286a4910c3a8578502c854c09b9ed79fd73f53b4d089435 \\
Pure Prop Address: & TMNf9aFM1UsRXJkDzFmAMwGHYvakMgcytbx \\
Theory Prop Id: & 70c25038dc2d795e7cd8a963a601a4e91c9ec37c890e90874b9e2ffe3758f817 \\
Theory Prop Address: & TMTYBnzfsAwJny6EQ3AaAXwwpjV8sXaQQm4
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_notin_tagged_Repl
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfogpgehjgogpfehbghghgfgegpfcffgahmg} ${\unicdcdadad \alpha Y .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\alpha}\unicdcdadjd{\{{\posthc{{y}}}|y \in {Y}\}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7f35b530bae383bd34f04f1aeb8e7f87cad1d17284bd4ad34bd07b9a9c0888ad \\
Pure Prop Address: & TMEiGdaBPeGmtidArg3tTV7mQKNB6Kjsv29 \\
Theory Prop Id: & e7ff3b50c4b94c722012bd9c9578bf3faa3faed0b7e0a6728349051f41a649fb \\
Theory Prop Address: & TMUFgGXyhtzJfcJ5jwFoArpbHCNJMjtGFEC
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgfemgehdhpf}
We define $\mname{SNoElts\_}$ to be ${\uniddcgcg \alpha .  {{\alpha}\unicdcdcdbg{\{{\posthc{{\beta}}}|\beta \in {\alpha}\}}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & c0ec73850ee5ffe522788630e90a685ec9dc80b04347c892d62880c5e108ba10 \\
Pure Object Address: & TMchcPtmntTsG9a2jnVbo3rvhd4YRFGdXd1 \\
Theory Object Id: & 6f0c364e1773eaa440ffcfc12f522a8c0532bdf6abfa9b8337d5e824777d1834 \\
Theory Object Address: & TMX8tC9smhsjcxum9MHG47kxXC2911bY8fh
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
SNoElts_mon
\end{verbatim}
\begin{theorem}\label{thm:dfoepgfemgehdhpfngpgog} ${\unicdcdadad \alpha \beta .  {{{\alpha} \subseteq {\beta}}\unicdbdjdcd{{{\mname{SNoElts\_}}~{\alpha}} \subseteq {{\mname{SNoElts\_}}~{\beta}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 49a08e4bc4f84602c91d1206ee5c208423e9081b680beb0711763d769896fcfb \\
Pure Prop Address: & TMcnJXLfgf6uG9ovz9CcN8AxtyhCcG7uj6R \\
Theory Prop Id: & 85961174f4e845853804252a1ed8497fdf7987feab6458860152819829435ac1 \\
Theory Prop Address: & TMYbW9UULsAfXGSQRuG31aqjM693gccqxqg
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgpf}
  We define $\mname{SNo\_}$ to be
  $${\uniddcgcg \alpha x .  {{{x} \subseteq {{\mname{SNoElts\_}}~{\alpha}}}\unicdcdcdhd{\unicdcdadad \beta \in {\alpha} .  {{{\mname{exactly1of2}}~{({{\posthc{{\beta}}} \in {x}})}}~{({{\beta} \in {x}})}}}}}$$
  of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 4ab7e4afd8b51df80f04ef3dd42ae08c530697f7926635e26c92eb55ae427224 \\
Pure Object Address: & TMWTZaSAkv4fbU4uw785MTxgHkscdn5nC4o \\
Theory Object Id: & c16a736947afa02fd21f8d698f4b28435c6d1a029515699f6dd4c54682d31685 \\
Theory Object Address: & TMQpDMmMJpF7k2EcsWrxASKz43UULkewFnB
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:afdfoepg}
We define $\mname{PSNo}$ to be ${\uniddcgcg \alpha p .  {{\{\beta \in {\alpha}|{{p}~{\beta}}\}}\unicdcdcdbg{\{{\posthc{{\beta}}}|\beta \in {\alpha}, {\preoh{{{p}~{\beta}}}}\}}}}$ of type ${{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{o}})}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 3657ae18f6f8c496bec0bc584e1a5f49a6ce4e6db74d7a924f85da1c10e2722d \\
Pure Object Address: & TMYCsiv2Uki9NzUmKkGH2kHd2zXGFWsYi5p \\
Theory Object Id: & 0a835174a0cf9cb97b20e29b98340a75a9c5f19db25004ae0688b25995ff83b3 \\
Theory Object Address: & TMTsRL2Kcwjhfick7fiiXD612CpaX6QFJea
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
PNoEq_PSNo
\end{verbatim}
\begin{theorem}\label{thm:afoepgfebhpfafdfoepg} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{{{\mname{{PNoEq\_}}}}~{{\alpha}}}}~{{({{\uniddcgcg \beta .  {{{{\beta}} \in {{{{{{\mname{{PSNo}}}}~{{\alpha}}}}~{{p}}}}}}}})}}}}~{{p}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c12e023a1bba6d9a9ec78087054b4057985c72eed826a9b312ba4f9091a65a61 \\
Pure Prop Address: & TMZtDRwJxRs7oFkrbsWnLiWufDfjc6F5M8R \\
Theory Prop Id: & 688518e3201b270ffc6ae7f6eaea9a64dca9d99cd02f4c980cb094834f164104 \\
Theory Prop Address: & TMWKeiYteNuQ4ETYrWy5zjJ7ErjwUTNfcGU
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_PSNo
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfafdfoepg} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{\mname{{SNo\_}}}}~{{\alpha}}}}~{{({{{{{{\mname{{PSNo}}}}~{{\alpha}}}}~{{p}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 96e4e2ec36857754b409849e1c10332c78d89197c28a888ac4de3c1e39dee14e \\
Pure Prop Address: & TMHEHzxt38dr1Sf22Q4XwK3oTJfyqBauuXx \\
Theory Prop Id: & f03db005e0c97b51d3b6d91f5cc5af548be05d0a792345091ee88d22daa20473 \\
Theory Prop Address: & TMUdwihfVPMkko4XmqQdHcyTjgzoGYZRGjh
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_PSNo_eta_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfafdfoepgpffgehbgpf} 
$$
\begin{array}{c}
{\unicdcdadad \alpha x .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNo\_}}}}~{{\alpha}}}}~{{x}}}}}{\unicdbdjdcd}{{{{x}}={{{{{{\mname{{PSNo}}}}~{{\alpha}}}}~{{({{\uniddcgcg \beta .  {{{{\beta}} \in {{x}}}}}})}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1642af79554e4cf7b9318de374b2c7edd186172454054d947f607438269c9496 \\
Pure Prop Address: & TMcuWpQkUmH1LMTde4kSNzc7YBDbA8sokbU \\
Theory Prop Id: & 630119c550dd8b5c2e0bd53e2ad889d4d08f72bc50e97a63bd942e63a6413115 \\
Theory Prop Address: & TMdU6L38rj7HbtERSXfWaxWZ7ZTCBb4pt8p
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepg}
$\mname{SNo}$ is the opaque object of type ${{\iota}\unicdbdjdcd{o}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 11faa7a742daf8e4f9aaf08e90b175467e22d0e6ad3ed089af1be90cfc17314b \\
Pure Object Address: & TMN8jh3ustcw3ZSbqUeiv8YSjeH79sfuWBi \\
Theory Object Id: & 116b7641893e1ad3998a7adf6c1fab6d1343fb60d499aef43ca12a4f847ef6b9 \\
Theory Object Address: & TMcKkMrp1PUTPmiJaXVAxaYrixRdqgEjZje
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
SNo_SNo
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfdfoepg} ${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad z .  {{{{\mname{SNo\_}}~{\alpha}}~{z}}\unicdbdjdcd{{\mname{SNo}}~{z}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 929b7ac3599669289a870d939e634944e175e96faf30cb744f95d10af1ed5d72 \\
Pure Prop Address: & TMMDuEgEyuT5E1BibeHE8pfR45YK1o1Bthr \\
Theory Prop Id: & a59341cc6c21bd7a5fd6bbf9d96c7141e1e9a5f9d341f307e50b304d6ed01584 \\
Theory Prop Address: & TMUJvhsT86hcmvoRoqef5N9GPieDsaTYJah
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgmefggh}
$\mname{SNoLev}$ is the opaque object of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 293b77d05dab711767d698fb4484aab2a884304256765be0733e6bd5348119e8 \\
Pure Object Address: & TMLgHPbDhEczHhzhujz1Y6647YGuRwkP7uG \\
Theory Object Id: & 83dce3cb89857056a3ed5cb650ae31e37721c49405fcf77a3b272f46d3ab6b80 \\
Theory Object Address: & TMUZTu2xUcdtVkLHYv2sJFX8Yj2S6Jd5jnk
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
SNoLev_uniq_Subq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpffhogjgbhpfdffhcgbh} 
$$
\begin{array}{c}
{\unicdcdadad x \alpha \beta .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{\mname{{ordinal}}}}~{{\beta}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNo\_}}}}~{{\alpha}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNo\_}}}}~{{\beta}}}}~{{x}}}}}{\unicdbdjdcd}{{{{\alpha}} \subseteq {{\beta}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 2ca114eb623f9aa0591aadc7c354900445384cf0262c752fa0430e62b230d79a \\
Pure Prop Address: & TMJjyz93md1dpAbsBqGeb8qeQUo8pB87vCg \\
Theory Prop Id: & 65843d7b872576b70cff22d1e596ee268a921435cd21f9de56a360ecd581bb9a \\
Theory Prop Address: & TMUMCHaN9FLoTQWs3Q4fEncBkHicwqi4FfL
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_uniq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpffhogjgbh} 
$$
\begin{array}{c}
{\unicdcdadad x \alpha \beta .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{\mname{{ordinal}}}}~{{\beta}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNo\_}}}}~{{\alpha}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNo\_}}}}~{{\beta}}}}~{{x}}}}}{\unicdbdjdcd}{{{{\alpha}}={{\beta}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 68f71dfd2001b27983e3c169b043d92f238faadc82a072f2d495cdc9761d2189 \\
Pure Prop Address: & TMaDQTErBA2zHWoN1hNMatRbHaH7eAJwMWF \\
Theory Prop Id: & 5d96d48026e74cd817ee90319b23cc2b42e20eba8d81f83f0edfcc2159f6ea5e \\
Theory Prop Address: & TMdvMVpBuYy5UApCwAKZdNQE9dn3JHxmPy6
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_prop
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpfahchpgah} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{\mname{{ordinal}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}\unicdcdcdhd{{{{{{\mname{{SNo\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}~{{x}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a8afccaac35cd67cae38ea5c7f8db5efcc00f46db4396c04d7693f267b847aa8 \\
Pure Prop Address: & TMEuZs9WhvFjxtstDDHiS5zYgaamp73ip3m \\
Theory Prop Id: & 21761c1d62638b04252039c756c82b2f54681ae14cef2e53b054c7eed1be1251 \\
Theory Prop Address: & TMTeCDaoMd1YLP8PGYRivc2WeuktskbT8SH
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_ordinal
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpfpgchegjgogbgmg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{\mname{ordinal}}~{({{\mname{SNoLev}}~{x}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7cacc5f1e879f627f484fd25c7b6c53a51432ebb794812d1bf85218be858870a \\
Pure Prop Address: & TMYpWCPBF4fDYL83uuTLHroYN9eaJFa1gjQ \\
Theory Prop Id: & f0e9f7da1d1eea44bc8152e3a3c9944f3397f3099eb97e82a027f58d2c354304 \\
Theory Prop Address: & TMNg3i1WfBeNE9GZ3VagfsKKYV6q3v2MqxJ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpf} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo\_}}~{({{\mname{SNoLev}}~{x}})}}~{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4a10339549425e6a1bcdd71c2ab61157802ad96e57ac87b6d61abeeb14663bc2 \\
Pure Prop Address: & TMF7HEEExZuLEM6qBULBQ44p5UsmzSwGq3s \\
Theory Prop Id: & 49cd5a80c0a8afed4364ca62e2e89621ac660e834b9bc67ddf46be5aeb6d8009 \\
Theory Prop Address: & TMHMibDhQbDodJRNsKTwFaztQh19uChMD8h
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_PSNo_eta
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfafdfoepgpffgehbg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{x}={{{\mname{PSNo}}~{({{\mname{SNoLev}}~{x}})}}~{({\uniddcgcg \beta .  {{\beta} \in {x}}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6ffefe7aa4ffb2d1b5711998134756f9e5e8326d515f3a9f165fa2d847ae71c8 \\
Pure Prop Address: & TMTaGgEBt5cPzRbi7CBcmpXvUfY5LgAtNjV \\
Theory Prop Id: & 8c78704c4a4036ae8fdbd50b6aa2091682a9b6cc08d150bb5d09f8689293ae7d \\
Theory Prop Address: & TMPZ6V2HX31zAKKRowAqxj1W7MhW2vtEkh3
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_PSNo
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpfafdfoepg} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{{\mname{{SNoLev}}}}~{{({{{{{{\mname{{PSNo}}}}~{{\alpha}}}}~{{p}}}})}}}}={{\alpha}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 07f1eb48a9b85d0d91b1644c1d205b7b5a1b4598605dbc92bcc702a5f6548159 \\
Pure Prop Address: & TMS4B9MsCny6DcgEj6Q5jyZtqGjyadqVrXb \\
Theory Prop Id: & 9c4890c1c16e11041619677ff7a9cab66327b15dc07aa9724197a3baf15864e4 \\
Theory Prop Address: & TMH875f6xnwhLVBj8FhdaBY27tPcA7tc93w
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_Subq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfdffhcgbh} 
$$
\begin{array}{c}
{\unicdcdadad x y .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoLev}}}}~{{x}}}} \subseteq {{{{\mname{{SNoLev}}}}~{{y}}}}}}}{\unicdbdjdcd}{{{({{\unicdcdadad \alpha \in {{{{\mname{{SNoLev}}}}~{{x}}}} .  {{{{{{\alpha}} \in {{x}}}}\unicdbdjded{{{{\alpha}} \in {{y}}}}}}}})}}}\\
{\unicdbdjdcd}{{{{x}} \subseteq {{y}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a244d79cdb92b7d27cf9364b054d3ab11f8fe1a80867c8aee4bfdbab426dc71b \\
Pure Prop Address: & TMZCRMtpxoXjbsEUAwQVnA3o3i8r9w9chR7 \\
Theory Prop Id: & a428aff2b59424a8834bd98dcf9cccb10f9ccd50a36aac8cdb05bdd1f8b0dc7b \\
Theory Prop Address: & TMVu9eEtBLYvjJeiHT4owg9NLx7ADETn241
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgfebhpf}
  We define $\mname{SNoEq\_}$ to be
  $${\uniddcgcg \alpha x y .  {{{{\mname{PNoEq\_}}~{\alpha}}~{({\uniddcgcg \beta .  {{\beta} \in {x}}})}}~{({\uniddcgcg \beta .  {{\beta} \in {y}}})}}}$$
  of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 5f11e279df04942220c983366e2a199b437dc705dac74495e76bc3123778dd86 \\
Pure Object Address: & TMHgy5CsvqeCa98Aq7nixfsuoKHSwuUheGS \\
Theory Object Id: & 12b5f61704c15020e2c024d9a3c4b44f386d2447a7bfd3a7ba2d9be7e534d79f \\
Theory Object Address: & TMPsS6FU8RDY8bj6bhgs9hNWme6t8mxkcHv
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
SNoEq_I
\end{verbatim}
\begin{theorem}\label{thm:dfoepgfebhpfje} ${\unicdcdadad \alpha x y .  {{({\unicdcdadad \beta \in {\alpha} .  {{{\beta} \in {x}}\unicdbdjded{{\beta} \in {y}}}})}\unicdbdjdcd{{{{\mname{SNoEq\_}}~{\alpha}}~{x}}~{y}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a718b1b74fe6e25f89cd15b443e1050c4627de6e63f7fc13aed6604a5dff174d \\
Pure Prop Address: & TMaoxpA3u5BRdKqSyxRArRdQKxBCCb4Ynfd \\
Theory Prop Id: & fdcb780aed4882ebdf83c46eeec00af5f3c06f91828ac6f34b8cd53807a3f3f0 \\
Theory Prop Address: & TMQV9XrhBpFPAX5ZadNCNzSub81tr3rR5pd
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoEq_E
\end{verbatim}
\begin{theorem}\label{thm:dfoepgfebhpffe} ${\unicdcdadad \alpha x y .  {{{{{\mname{SNoEq\_}}~{\alpha}}~{x}}~{y}}\unicdbdjdcd{\unicdcdadad \beta \in {\alpha} .  {{{\beta} \in {x}}\unicdbdjded{{\beta} \in {y}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 322284f9b12d82137cb4547584023ca4089ff501f8b46d31017d6644359f67bc \\
Pure Prop Address: & TMUx62argfoVk7QVYzx9FBiQWTGGzPgSkMr \\
Theory Prop Id: & 76c08c80b9ae86c43f88454225e5c2d37189dab1f7a4aaaab54885b6d1d584d5 \\
Theory Prop Address: & TMcZ243SUoVmTGmz8GBWnNRSGSoQLcSekNM
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoEq_E1
\end{verbatim}
\begin{theorem}\label{thm:dfoepgfebhpffebd} ${\unicdcdadad \alpha x y .  {{{{{\mname{SNoEq\_}}~{\alpha}}~{x}}~{y}}\unicdbdjdcd{\unicdcdadad \beta \in {\alpha} .  {{{\beta} \in {x}}\unicdbdjdcd{{\beta} \in {y}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & bc569f4fe8f8babfb01e30d1ce5330c3cd167c3a34a4d7f6865bbcb63ff0c246 \\
Pure Prop Address: & TMcpXWEGwJyBfkywuB1xBLvouHwBSaFs2Lu \\
Theory Prop Id: & c702b67d3382793ff6831ae5d9948c42fcdf9959f67e4f34b70f105a4fa627c0 \\
Theory Prop Address: & TMKUKLNUuNym8woqSCRiDcMxAfsCb1eUBk1
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoEq_E2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgfebhpffecd} ${\unicdcdadad \alpha x y .  {{{{{\mname{SNoEq\_}}~{\alpha}}~{x}}~{y}}\unicdbdjdcd{\unicdcdadad \beta \in {\alpha} .  {{{\beta} \in {y}}\unicdbdjdcd{{\beta} \in {x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 04c55ef7be9110a6c332fe11ca098564ded45c1ae7f131199dc8a18eff3f728a \\
Pure Prop Address: & TMa2w3rdW3KJYQkLRhjwBtZCzeLY2CsSzUt \\
Theory Prop Id: & 06351353e81c1d6cb1d4bd1b327570332781fae5e00a0975bce060a995406d00 \\
Theory Prop Address: & TMTis8K7byF4GTjz4MpYG9BkyoJCzkUAJyW
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoEq_antimon_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgfebhpfbgogehjgngpgogpf} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad \beta \in }{{{\alpha}}}{ .  }{\unicdcdadad x y .  }{{{{{{{{{\mname{{SNoEq\_}}}}~{{\alpha}}}}~{{x}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{SNoEq\_}}}}~{{\beta}}}}~{{x}}}}~{{y}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8d4936ed8e712c1dc231fb4b91a23637000f913b108f46cd2edc43f5e8726ba6 \\
Pure Prop Address: & TMFyyANM4SzwuZGwLMdhVdAWuiVUHstHQug \\
Theory Prop Id: & 95a70a9933c8b7e36b75adaa97b483d7f470c88d64871a9b077468393a7cee44 \\
Theory Prop Address: & TMG63bj4zRBfeAkNC86WKZ7gtUpSwCSprJR
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_eq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgbh} 
$$
\begin{array}{c}
{\unicdcdadad x y .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoLev}}}}~{{x}}}}={{{{\mname{{SNoLev}}}}~{{y}}}}}}}{\unicdbdjdcd}\\
{{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}~{{x}}}}~{{y}}}}}{\unicdbdjdcd}{{{{x}}={{y}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b08bdc8e0533c6d335bfaefe372cf3eae5a6596181f4279fbee1bbbb96bd58d2 \\
Pure Prop Address: & TMMawfQkhohvkcRg13ZBDTRwn1DgFdvbZN8 \\
Theory Prop Id: & bf4ac3765cc4e77e69dfed6b06b79b2a135de381ed468aff0717b55ecd997219 \\
Theory Prop Address: & TMXZ1YaK7uq7CL6xjMR9UpcJae85o8Qg6an
\end{tabular}
\end{center}

\end{theorem}
Let $ctag : {{\iota}\unicdbdjdcd{\iota}}$ be ${\uniddcgcg \alpha .  {{{\mname{SetAdjoin}}~{\alpha}}~{\{{2}\}}}}$.
{\bf{Notation.}} We use $\posthchc{-}$ as a postfix operator  corresponding to applying term ${ctag}$.
\begin{verbatim}
ctagged_not_ordinal
\end{verbatim}
\begin{theorem}\label{thm:dgehbghghgfgegpfogpgehpfpgchegjgogbgmg} ${\unicdcdadad y .  {\preoh{{{\mname{ordinal}}~{({\posthchc{{y}}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 06ba78058810b2a66de61b42a989e96b9faf1830fe88331433cb796ec5e9efba \\
Pure Prop Address: & TMSJ67Y6zbYZXw4KipNfSDFXS7nCowqPsfv \\
Theory Prop Id: & 762f4db9b4131082faa31f45fdf77ff2ecd9cada9a2b8be09a7c87338296ffaa \\
Theory Prop Address: & TMUUZwa9YUDXipGjVpau4QY3LpPAUNonrus
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ctagged_notin_ordinal
\end{verbatim}
\begin{theorem}\label{thm:dgehbghghgfgegpfogpgehjgogpfpgchegjgogbgmg} ${\unicdcdadad \alpha y .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{({\posthchc{{y}}})}\unicdcdadjd{\alpha}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a6d7ced029db2bf647e1bcf629a266dc8aa5350b597ac826ee1499c9488ef153 \\
Pure Prop Address: & TMJtxbUm4vTnZsFYwmfoJXkdyU9Pa5iDTMG \\
Theory Prop Id: & 33eb91696d423e27f32912360405bcff4635722555c6e964c127195a36bc3b46 \\
Theory Prop Address: & TMU4pDKct3EFb68TQMdmzrbHxh4jXHXLAx4
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
Sing2_notin_SingSing1
\end{verbatim}
\begin{theorem}\label{thm:dfjgoghgcdpfogpgehjgogpfdfjgoghgdfjgoghgbd} ${{\{{2}\}}\unicdcdadjd{\{{\{{1}\}}\}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f44be8cc1b2eabcd18be5c1af7cc7c118087af9d39e568bff28319aa056768c9 \\
Pure Prop Address: & TMYYY1TCNHPhoU7ioo1yEjqPen8SD381J2H \\
Theory Prop Id: & 4504e136e5be3235264222f1b4fc92b8c82895870a0d5af8fb0ca383d0aa0b93 \\
Theory Prop Address: & TMXydjLtD7zAbYMVdsbyZnBWtZSoHs13VUk
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ctagged_notin_SNo
\end{verbatim}
\begin{theorem}\label{thm:dgehbghghgfgegpfogpgehjgogpfdfoepg} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{({\posthchc{{y}}})}\unicdcdadjd{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e87bab002c13fd4c8f45d1fab441c351590059c0546ef7cb97b2f419eded12bb \\
Pure Prop Address: & TMLi9ZahvNViXr94ACKuVosVxgffZSXBaQh \\
Theory Prop Id: & fb05c7f6bcb781bf45159e3631adfc011ed3dcf028713a0a5e65ffa8442e7de6 \\
Theory Prop Address: & TMdvxEZ5qYtgLAENGC1mYi7PPenMSpHbbbG
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ctagged_eqE_eq
\end{verbatim}
\begin{theorem}\label{thm:dgehbghghgfgegpffgbhfepffgbh} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{\unicdcdadad u \in {x} .  {\unicdcdadad v \in {y} .  {{{\posthchc{{u}}}={\posthchc{{v}}}}\unicdbdjdcd{{u}={v}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9635c2246bd5ce564e4af804699383bd45289a4ce4fef5acfb843ea16be935ac \\
Pure Prop Address: & TMWuHzjRejZ3vCngDLVxyx22KfXifdUFgrX \\
Theory Prop Id: & 7a7dec8b0cb27354ad3358e329e40af18501227e17ce9a3c573fb2ced3933325 \\
Theory Prop Address: & TMFudCkTrZ8qDZwsaKBW9Qsuj6AjLeAqYtM
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgpfahbgjgch}
We define $\mname{SNo\_pair}$ to be ${\uniddcgcg x y .  {{x}\unicdcdcdbg{\{{\posthchc{{u}}}|u \in {y}\}}}}$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 0c801be26da5c0527e04f5b1962551a552dab2d2643327b86105925953cf3b06 \\
Pure Object Address: & TMQqNXnBZnkKSe8hed4os6KPYmLwpS71Jw7 \\
Theory Object Id: & 88bcd797eb0ac50799968a11f7c23c8ed718c981b7a3bb422ca738771da70d49 \\
Theory Object Address: & TMS1SBjNkN4Fv979vSCovhJ3Hm5LmNC8vSP
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
SNo_pair_prop_1
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfahbgjgchpfahchpgahpfbd} 
$$
\begin{array}{c}
{\unicdcdadad x1 y1 x2 y2 .  }{{{{{\mname{{SNo}}}}~{{x1}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{x2}}}}}{\unicdbdjdcd}{{{{{{{{{\mname{{SNo\_pair}}}}~{{x1}}}}~{{y1}}}}={{{{{{\mname{{SNo\_pair}}}}~{{x2}}}}~{{y2}}}}}}}{\unicdbdjdcd}\\
{{{{x1}}={{x2}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 55e11f2068e4bac205e7f81aa76c24ddba355ab4507585cc4d8d2d2f4087857f \\
Pure Prop Address: & TMY6PgwrmwyiN6arDm1Yhg5JrTC3jXy8Z7g \\
Theory Prop Id: & f423640149097053118cefcbb2f2260329493536ad3ae58e47efbdc5cf96a62f \\
Theory Prop Address: & TMabWw9KKwUpjo1s2ey6gDkv7hUyHidq12D
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_pair_prop_2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfahbgjgchpfahchpgahpfcd} 
$$
\begin{array}{c}
{\unicdcdadad x1 y1 x2 y2 .  }{{{{{\mname{{SNo}}}}~{{x1}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y1}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{x2}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y2}}}}}{\unicdbdjdcd}\\
{{{{{{{{{\mname{{SNo\_pair}}}}~{{x1}}}}~{{y1}}}}={{{{{{\mname{{SNo\_pair}}}}~{{x2}}}}~{{y2}}}}}}}{\unicdbdjdcd}{{{{y1}}={{y2}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0bfff99d8cefbebcd6db7207b1609c8e7c6338fc2381f6c937e65ef66435fa02 \\
Pure Prop Address: & TMZQx8qdiMuWPMgQr92RnVZJPMwYU6eRoDU \\
Theory Prop Id: & cc5febed625436e972c38a7e71012d4d0459f0651c13dba74988508db075847c \\
Theory Prop Address: & TMboEtYmU1rhLndeftmE1xvAXa12uenk52T
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_pair_prop
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfahbgjgchpfahchpgah} 
$$
\begin{array}{c}
{\unicdcdadad x1 y1 x2 y2 .  }{{{{{\mname{{SNo}}}}~{{x1}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y1}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{x2}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y2}}}}}{\unicdbdjdcd}\\
{{{{{{{{{\mname{{SNo\_pair}}}}~{{x1}}}}~{{y1}}}}={{{{{{\mname{{SNo\_pair}}}}~{{x2}}}}~{{y2}}}}}}}{\unicdbdjdcd}{{{{{{x1}}={{x2}}}}\unicdcdcdhd{{{{y1}}={{y2}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0c1fab5088198fbd78c0308f1d0a7d3e7a9067d7ed242c17489a18a01a610489 \\
Pure Prop Address: & TMdh56g5Z6tVNavL7VJHdhuafzTH1Pmfs82 \\
Theory Prop Id: & 9f8e820a171395b7e1ea5696b5ba61536ab0a1efd23d479c1ad3f329425325a3 \\
Theory Prop Address: & TMYoD7ZXdXJGFLDcspavQztKyvhuDPUyktz
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_pair_0
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfahbgjgchpfad} ${\unicdcdadad x .  {{{{\mname{SNo\_pair}}~{x}}~{0}}={x}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a24e9ca908c70bad12f23288cfeadb94cf8d4a42e36b819d9a269d2083a09ee7 \\
Pure Prop Address: & TMLieswQhd4ZG8isZSgFpxUfMX6vYzwXue7 \\
Theory Prop Id: & fe382ec7ec6028476f54901b6bab3e279d994b93eb8d072db22da7f5d6c039d6 \\
Theory Prop Address: & TMZuZFbfyTNy3vBnkV22aEUTbRJMeouwCtc
\end{tabular}
\end{center}

\end{theorem}
\section{Surreal Numbers as Sets II} % TaggedSets
\begin{definition}\label{def:dfoepgmeeh}
  We define $\mname{SNoLt}$ to be
  $${\uniddcgcg x y .  {{{{{\mname{PNoLt}}~{({{\mname{SNoLev}}~{x}})}}~{({\uniddcgcg \beta .  {{\beta} \in {x}}})}}~{({{\mname{SNoLev}}~{y}})}}~{({\uniddcgcg \beta .  {{\beta} \in {y}}})}}}$$
  of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 46e7ed0ee512360f08f5e5f9fc40a934ff27cfd0c79d3c2384e6fb16d461bd95 \\
Pure Object Address: & TMQeBApbcRbbcKWZqVjr7EnqQ7iX7wkV5cx \\
Theory Object Id: & 8c8aa9c7cd39e813a521dcca067bdc27aa59721372e00112d3a56e0af7f1c820 \\
Theory Object Address: & TMKc4rFp1uKxLq8fj5pXFZziozMvupbPz34
\end{tabular}
\end{center}

\end{definition}
{\bf{Notation.}} We use $\infmd$ as an infix operator  corresponding to applying term ${\mname{SNoLt}}$.
\begin{definition}\label{def:dfoepgmefg}
  We define $\mname{SNoLe}$ to be
  $${\uniddcgcg x y .  {{{{{\mname{PNoLe}}~{({{\mname{SNoLev}}~{x}})}}~{({\uniddcgcg \beta .  {{\beta} \in {x}}})}}~{({{\mname{SNoLev}}~{y}})}}~{({\uniddcgcg \beta .  {{\beta} \in {y}}})}}}$$
  of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & ddf7d378c4df6fcdf73e416f8d4c08965e38e50abe1463a0312048d3dd7ac7e9 \\
Pure Object Address: & TMKo92SvXPoMheM1GrcBcnHBZDewaXgqJwX \\
Theory Object Id: & 54f405ff1c52a47311b5896cb2435cbca5240391e95a4473bb33d4f389ef4b27 \\
Theory Object Address: & TMcgcuHLpN7pQTJ2oMAMaweT9JaE3NrXnSu
\end{tabular}
\end{center}

\end{definition}
{\bf{Notation.}} We use $\infmdnd$ as an infix operator  corresponding to applying term ${\mname{SNoLe}}$.
\begin{verbatim}
SNoLtLe
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehmefg} ${\unicdcdadad x y .  {{{x}<{y}}\unicdbdjdcd{{x}\unicdcdgded{y}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c0fef1fff2b54737a2955837a95cd2fa4a51eaeab8bdcd197730a7dc32fdaa00 \\
Pure Prop Address: & TMcpCGuF3ij7f7Q9otaxmn58tuxgzTAEELE \\
Theory Prop Id: & f09833f8c9ddb44132bc696213b112e392eb8293700f8a6493ef998a11ef5bf7 \\
Theory Prop Address: & TMWx22uWdLVXSdBruTx9wkh1hYKRqrGaxYL
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLeE
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgfe} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}\unicdcdgded{y}}\unicdbdjdcd{{{x}<{y}}\unicdcdcdid{{x}={y}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1c9c62734eefe76ad0f740a1f3c250414dfb8d30fa252d1ca61a69f245fe533c \\
Pure Prop Address: & TMKZjxSSN6F5jwo8R4FwwTKANQJT8wVY6Zq \\
Theory Prop Id: & 23cc476eb8b886081aab3ea46dc0b4ab0cbfb0a43df47487eade402bb40d21d1 \\
Theory Prop Address: & TMR8soSdPW2WyGNYANuCH2tdDGwy3XK3ZZX
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoEq_ref_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgfebhpfchfgggpf} ${\unicdcdadad \alpha x .  {{{{\mname{SNoEq\_}}~{\alpha}}~{x}}~{x}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 259cc3291296e3db90ffb96d963d0421fb6016e7a41218d81b92cd846193aa78 \\
Pure Prop Address: & TMGJ1mcTbnMvajkoJpACBxXXWZPaQ116hPq \\
Theory Prop Id: & 92ef044c4eddbbb417dc88765739163aa0b710f5b5646ffae3c372cffa7a575a \\
Theory Prop Address: & TMWvCZMB6S5XhCvHow2Zc5MwWv32acftDJc
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoEq_sym_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgfebhpfdhjhngpf} ${\unicdcdadad \alpha x y .  {{{{{\mname{SNoEq\_}}~{\alpha}}~{x}}~{y}}\unicdbdjdcd{{{{\mname{SNoEq\_}}~{\alpha}}~{y}}~{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f8c61d4617dba1594bda7ef1b25d222315df97512407b3dc4d77c1e117f5ffae \\
Pure Prop Address: & TMYtRm1GvHqLhau3uVcMnTfMYn5u2YmwqHi \\
Theory Prop Id: & f756d24db9f6575a09cab35daef6bf28fa9b1b0101f4d013fedfef1e8c0f7741 \\
Theory Prop Address: & TMZPd22JbRT6sSfR5cHZF68img19UYG8kuM
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoEq_tra_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgfebhpfehchbgpf} 
$$
\begin{array}{c}
{\unicdcdadad \alpha x y z .  }{{{{{{{{{\mname{{SNoEq\_}}}}~{{\alpha}}}}~{{x}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{{{{{\mname{{SNoEq\_}}}}~{{\alpha}}}}~{{y}}}}~{{z}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{SNoEq\_}}}}~{{\alpha}}}}~{{x}}}}~{{z}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0fff96e9fa087a01b4e627c5a43d8f712211ef77acbd62778d0d5fbdc9a2c11e \\
Pure Prop Address: & TMGUGWpuTgNVZaVHrjqqx3B3dvveuU3xb1w \\
Theory Prop Id: & 246d040775eff6f1c7748648c0c49f13cbd822ffecf62018af28e16107c8db95 \\
Theory Prop Address: & TMKhJm34Nbo1ihoXZ31EHQfG87mFoxCD2xF
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLtE
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehfe} 
$$
\begin{array}{c}
{\unicdcdadad x y .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{x}}<{{y}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{o}}}{ .  }\\
(\unicdcdadad z .  {{{{\mname{{SNo}}}}~{{z}}}}\unicdbdjdcd{{{{{{\mname{{SNoLev}}}}~{{z}}}} \in {{{{{{\mname{{SNoLev}}}}~{{x}}}}\unicdcdcdjd{{{{\mname{{SNoLev}}}}~{{y}}}}}}}}\unicdbdjdcd\\
{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{z}}}})}}}}~{{z}}}}~{{x}}}}\unicdbdjdcd{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{z}}}})}}}}~{{z}}}}~{{y}}}}\unicdbdjdcd{{{{x}}<{{z}}}}\unicdbdjdcd\\
{{{{{{z}}<{{y}}}}\unicdbdjdcd{{{{{{{{\mname{{SNoLev}}}}~{{z}}}}\unicdcdadjd{{x}}}}\unicdbdjdcd{{{{{{{{\mname{{SNoLev}}}}~{{z}}}} \in {{y}}}}\unicdbdjdcd{{p}}}}}}}})\\
{\unicdbdjdcd}\\
{{{({{{{{{{{\mname{{SNoLev}}}}~{{x}}}} \in {{{{\mname{{SNoLev}}}}~{{y}}}}}}\unicdbdjdcd{{{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}~{{x}}}}~{{y}}}}\unicdbdjdcd{{{{{{{{\mname{{SNoLev}}}}~{{x}}}} \in {{y}}}}\unicdbdjdcd{{p}}}}}}}})}}}\\
{\unicdbdjdcd}\\
{{{({{{{{{{{\mname{{SNoLev}}}}~{{y}}}} \in {{{{\mname{{SNoLev}}}}~{{x}}}}}}\unicdbdjdcd{{{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}}~{{x}}}}~{{y}}}}\unicdbdjdcd{{{{{{{{\mname{{SNoLev}}}}~{{y}}}}\unicdcdadjd{{x}}}}\unicdbdjdcd{{p}}}}}}}})}}}\\
{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 958711d28833ece2b40c3f3b22c5429d43fda518512b7e6cd75b687901aa585c \\
Pure Prop Address: & TMQaPacdRra3VaMe2NYNa6hUY8TFNJ1no3E \\
Theory Prop Id: & d12fc61e3017695d6fd60f1d88de8697d1f3b34fae6748b05bc36f7361185ea5 \\
Theory Prop Address: & TMScxkqv8rKobWueAxfgzCcwhkfhaGV9qHT
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLtI2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehjecd} 
$$
\begin{array}{c}
{\unicdcdadad x y .  }{{{{{{{\mname{{SNoLev}}}}~{{x}}}} \in {{{{\mname{{SNoLev}}}}~{{y}}}}}}}{\unicdbdjdcd}{{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}~{{x}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoLev}}}}~{{x}}}} \in {{y}}}}}\\
{\unicdbdjdcd}{{{{x}}<{{y}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 282b0114a7969ac0a9f2f7d7736f7948b656946dfe0bd61474b62b3399f08148 \\
Pure Prop Address: & TMSSV4UzJfz3EizcWCjQwJdMaAvo7dQx3od \\
Theory Prop Id: & 82b82caa129216de39cbf73134cd56db7ca51386bfea22d4c8ddd6c051496ed1 \\
Theory Prop Address: & TMTsvbaNTUL6yQqdnrt1ngGJ6yz18T2WJx7
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLtI3
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehjedd} 
$$
\begin{array}{c}
{\unicdcdadad x y .  }{{{{{{{\mname{{SNoLev}}}}~{{y}}}} \in {{{{\mname{{SNoLev}}}}~{{x}}}}}}}{\unicdbdjdcd}{{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}}~{{x}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoLev}}}}~{{y}}}}\unicdcdadjd{{x}}}}}\\
{\unicdbdjdcd}{{{{x}}<{{y}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 006afc7cf0f802618dbec43d665276d0ee81d35772edbf373b3bd651342a9d37 \\
Pure Prop Address: & TMJ4VDCxrxGVAF5fK26vap4HDavQoHRMJ4m \\
Theory Prop Id: & 938f67cc578e6abe82c5585824484d30503f3bcc5e3689cb6de279568d894afc \\
Theory Prop Address: & TMX95fsRv9YXCLYg3BLQgUEsosPv95mRaVf
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLt_irref
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehpfjgchchfggg} ${\unicdcdadad x .  {\preoh{{{{\mname{SNoLt}}~{x}}~{x}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d5ecf97e33d758999475a13ee895246c6ae7bc234feff76c2354f56f7b624826 \\
Pure Prop Address: & TMbVeK3yqN3QTUGSqR22RZ7ufquPqAg75TV \\
Theory Prop Id: & 06fac5ec34e79c463fc32f8c6ed17874daafb68995b87ca166ab687f4266c6e8 \\
Theory Prop Address: & TMFNgRdNx48U2iQGe5ump9qwyA5gisWDyYk
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLt_trichotomy_or
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehpfehchjgdgigpgehpgngjhpfpgch} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{{x}<{y}}\unicdcdcdid{{x}={y}}}\unicdcdcdid{{y}<{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 35a4c8a73fc7c1c7cbab4f63f7e92de7f75eec2e52fca95b9fd47f0ab2de7ae7 \\
Pure Prop Address: & TMdfGBU7nCaoeRDa8DYFzE35sW9rF2BaYJT \\
Theory Prop Id: & 8ad056deb63dd86dc6b1c668c0e389f14827e04273dcb3e1a8d1c1e74bad1877 \\
Theory Prop Address: & TMdykwY9bimsgzdU8rQy52x82fcLDNP8gqm
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLt_trichotomy_or_impred
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehpfehchjgdgigpgehpgngjhpfpgchpfjgngahchfgeg} 
$$
\begin{array}{c}
{\unicdcdadad x y .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{o}}}{ .  }{{{({{{{{{x}}<{{y}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{{{({{{{{{x}}={{y}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{{{({{{{{{y}}<{{x}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ba04ef155f5e47607fdb0216fb6a06340f5cd9616eb6c817260c9f387b62270a \\
Pure Prop Address: & TMV8mPnyLgKgikuN7NsNsQJao2G8A2ya29W \\
Theory Prop Id: & 7588e3b81e8c26b5bdfa83b3a50190f9f22e3e7c278731520d46b07ea230bf45 \\
Theory Prop Address: & TMYEfgRm1btK3489ffptw9arr8NJ1tSuHYr
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLt_tra
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehpfehchbg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}<{y}}\unicdbdjdcd{{{y}<{z}}\unicdbdjdcd{{x}<{z}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ac0e241054ec1b90968f95ad1b556eec114498a58651d565ae4cac1b019867c7 \\
Pure Prop Address: & TMddkASjPFXDxWgZwPsWeWLvopbdDModThm \\
Theory Prop Id: & ccfdf7a1275e7ae74a5f6ecc7d8b2def5778c8e9002abdb744c67fd331be61e1 \\
Theory Prop Address: & TML6qNhmMdQ9xqiz61XAszuyuyC6VGsunnR
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLe_ref
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgpfchfggg} ${\unicdcdadad x .  {{{\mname{SNoLe}}~{x}}~{x}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c940c82ce0e40faca616524909eeb5303ed083f5290e54d8054cfbae1cf0ec2f \\
Pure Prop Address: & TMWbEJ22hQKnoK2eVaoyNrJEUXEkgbcRBJf \\
Theory Prop Id: & 83bd2d49035b19f0998a95f22664723ce6d3c960022e6125f8249aa45f2b1292 \\
Theory Prop Address: & TMKJtJGg7gcLDxHPUbGWPcUfzbs1mmF3Wki
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLe_antisym
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgpfbgogehjgdhjhng} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}\unicdcdgded{y}}\unicdbdjdcd{{{y}\unicdcdgded{x}}\unicdbdjdcd{{x}={y}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f9f7571d97142cd27d147c350951113898e84e19eedaf980f36ed322766e0514 \\
Pure Prop Address: & TMdEqGkU3vjpdRYetepi2YSrbnZ5VjCxYuZ \\
Theory Prop Id: & 3abee4eacee94652f2f428c3dd42de2ad807637c433d8c2c868a6509d7a6427a \\
Theory Prop Address: & TMMewh6x1J5khDBzMVayDe8pgeEGnJt2fPL
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLtLe_tra
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehmefgpfehchbg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}<{y}}\unicdbdjdcd{{{y}\unicdcdgded{z}}\unicdbdjdcd{{x}<{z}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3bc66187a3c8c6c503fb4e0c28121c6f7b0e9fdb129f4989a4e877bda7ea2331 \\
Pure Prop Address: & TMFeQ529o5NY2zuHeApHyghETBCcdzFrerN \\
Theory Prop Id: & ba8545b12dea14cf235ae9007eb85c28adfad81e6abd4a3c0afdc1263e529e5d \\
Theory Prop Address: & TMWXyUVVCVjMxZQeS5PSqxmnt47eXXrnZjj
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLeLt_tra
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgmeehpfehchbg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}\unicdcdgded{y}}\unicdbdjdcd{{{y}<{z}}\unicdbdjdcd{{x}<{z}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b2178f178198787e74b3d9f71600d68ffeef7e217a11d962036dfd3e08b75293 \\
Pure Prop Address: & TMLec6uaS4DS4VwQAcpcPM3YPb74q9SMcDB \\
Theory Prop Id: & 4a77a2a749de106eaac82a7c1e466ed95de5fc489f1820e8b49646e6b03c8d35 \\
Theory Prop Address: & TMK9rwT18dhoTbiBXjrvgGExGkmYmRnFyZK
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLe_tra
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgpfehchbg} ${\unicdcdadad x y z .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{\mname{SNo}}~{z}}\unicdbdjdcd{{{x}\unicdcdgded{y}}\unicdbdjdcd{{{y}\unicdcdgded{z}}\unicdbdjdcd{{x}\unicdcdgded{z}}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a0c5c60c2ceffb86861009ffeca522474ef473926e20770226f32c90efdc18d0 \\
Pure Prop Address: & TMKoicgHRM8yShdwoBNSAh9MEvz7q8PtkM9 \\
Theory Prop Id: & 0319892cfab3f54bb914dbb9f14645402420d106668b7793a5f85e69ecca72fc \\
Theory Prop Address: & TMXd83ANWfT4zya3nqhmvzCfE2ez1NuqTf3
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLtLe_or
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehmefgpfpgch} ${\unicdcdadad x y .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo}}~{y}}\unicdbdjdcd{{{x}<{y}}\unicdcdcdid{{y}\unicdcdgded{x}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f595ac154b5f6c53eca4ea196af34004c4273a448817a2233791468da205e94a \\
Pure Prop Address: & TMQDjtGFE58mqHck4JTyF12SPbWSzrFAjzw \\
Theory Prop Id: & 8965d2d632b616ea19510993aaf5d59d9c47af2a7d76be6d00d0b436c1309dc4 \\
Theory Prop Address: & TMXGgukiEQFsFuC6uSDXeRYCUnqF5inCRTa
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLt_PSNo_PNoLt
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehpfafdfoepgpfafoepgmeeh} 
$$
\begin{array}{c}
{\unicdcdadad \alpha \beta .  }{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{\mname{{ordinal}}}}~{{\beta}}}}}{\unicdbdjdcd}{{{{{{{{{\mname{{PSNo}}}}~{{\alpha}}}}~{{p}}}}<{{{{{{\mname{{PSNo}}}}~{{\beta}}}}~{{q}}}}}}}{\unicdbdjdcd}\\
{{{{{{{{{{\mname{{PNoLt}}}}~{{\alpha}}}}~{{p}}}}~{{\beta}}}}~{{q}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6e122de5852708a5ca608843957b6c9ff2dae5ef5222ddf994aad35720219c8f \\
Pure Prop Address: & TMG8BGEKrgDCqPJMuPr9e9EppTdzunca7fq \\
Theory Prop Id: & 535161c1205ef5061ddcd8bc9186161d3eacd8b5c97edac1dfa870877bb532ba \\
Theory Prop Address: & TMNDvRwj9viGBk1WZotrsPW535EA9L5CVb6
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
PNoLt_SNoLt_PSNo
\end{verbatim}
\begin{theorem}\label{thm:afoepgmeehpfdfoepgmeehpfafdfoepg} 
$$
\begin{array}{c}
{\unicdcdadad \alpha \beta .  }{\unicdcdadad p q : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{\mname{{ordinal}}}}~{{\beta}}}}}{\unicdbdjdcd}{{{{{{{{{{{\mname{{PNoLt}}}}~{{\alpha}}}}~{{p}}}}~{{\beta}}}}~{{q}}}}}{\unicdbdjdcd}\\
{{{{{{{{\mname{{PSNo}}}}~{{\alpha}}}}~{{p}}}}<{{{{{{\mname{{PSNo}}}}~{{\beta}}}}~{{q}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 11b6ed978d93a259a246a64548a3100d22b113d8b2de70d4392bb571d4078f73 \\
Pure Prop Address: & TMXkNqDx5R69XDH3RVS3UbfdKQwDetvxzFa \\
Theory Prop Id: & 6196d926339459b6e115949379593dc6e6a3f5922f1b60dd11abdd0193525e46 \\
Theory Prop Address: & TMSzfRKTbTeWkajRMYPfF6mjUqxx7M2QY5i
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgdefheh}
  We define $\mname{SNoCut}$ to be
  $$
  \begin{array}{c}
    \uniddcgcg L R .  {\mname{PSNo}}~\\
      {({{{\mname{PNo\_bd}}~{({\uniddcgcg \alpha p .  {{{\mname{ordinal}}~{\alpha}}\unicdcdcdhd{{{{\mname{PSNo}}~{\alpha}}~{p}} \in {L}}}})}}~{({\uniddcgcg \alpha p .  {{{\mname{ordinal}}~{\alpha}}\unicdcdcdhd{{{{\mname{PSNo}}~{\alpha}}~{p}} \in {R}}}})}})}~\\
               {({{{\mname{PNo\_pred}}~{({\uniddcgcg \alpha p .  {{{\mname{ordinal}}~{\alpha}}\unicdcdcdhd{{{{\mname{PSNo}}~{\alpha}}~{p}} \in {L}}}})}}~{({\uniddcgcg \alpha p .  {{{\mname{ordinal}}~{\alpha}}\unicdcdcdhd{{{{\mname{PSNo}}~{\alpha}}~{p}} \in {R}}}})}})}
  \end{array}
  $$ of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & ec849efeaf83b2fcdbc828ebb9af38820604ea420adf2ef07bb54a73d0fcb75b \\
Pure Object Address: & TMRuDcB8ezdcHgdFjNy68Wt7HG9e2X1bxLT \\
Theory Object Id: & 295bc4535cc47135a06fa73d8ca9d4df8cc9e2d68a2a449df25a754b7e9dc1a9 \\
Theory Object Address: & TMP7d7RzZhj9qAvayw3UXZSiTmHLbd188MH
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:dfoepgdefhehaf}
  We define $\mname{SNoCutP}$ to be
  $${\uniddcgcg L R .  {{{({\unicdcdadad x \in {L} .  {{\mname{SNo}}~{x}}})}\unicdcdcdhd{({\unicdcdadad y \in {R} .  {{\mname{SNo}}~{y}}})}}\unicdcdcdhd{({\unicdcdadad x \in {L} .  {\unicdcdadad y \in {R} .  {{x}<{y}}}})}}}$$
  of type ${{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{o}}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & c083d829a4633f1bc9aeab80859255a8d126d7c6824bf5fd520d6daabfbbe976 \\
Pure Object Address: & TMdYBKYErUYWZiKgnTvgUffZxEVBUVDaLvh \\
Theory Object Id: & 775127720c53377db4bdf359fa2ddfb72fd332c9614eaf97b7db372fc3b6837a \\
Theory Object Address: & TMF5e4Sx4fER7UgD1xGTzVfv1apiMjDYdoK
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
SNoCutP_SNoCut
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfdfoepgdefheh} 
$$
\begin{array}{c}
{\unicdcdadad L R .  }{{{{{{{\mname{{SNoCutP}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}\\
{{{{\mname{{SNo}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}}\unicdcdcdhd\\
{{{{\mname{{SNoLev}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}} \in\\
{{{{\mname{{ordsucc}}}}~{{({{{{({{\unicdcddedd _{{x \in {{L}}}}{{\mname{{ordsucc}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}})}}\unicdcdcdbg{{({{\unicdcddedd _{{y \in {{R}}}}{{\mname{{ordsucc}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}})}}}})}}}}\unicdcdcdhd\\
{{({{\unicdcdadad x \in {{L}} .  {{{{x}}<{{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}}}}}})}}\unicdcdcdhd\\
{{({{\unicdcdadad y \in {{R}} .  {{{{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}}<{{y}}}}}})}}\unicdcdcdhd\\
(\unicdcdadad z .  {{{{\mname{{SNo}}}}~{{z}}}}\unicdbdjdcd
{{({{\unicdcdadad x \in {{L}} .  {{{{x}}<{{z}}}}}})}}\unicdbdjdcd
{{({{\unicdcdadad y \in {{R}} .  {{{{z}}<{{y}}}}}})}}\unicdbdjdcd\\
{{{{{{\mname{{SNoLev}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}} \subseteq {{{{\mname{{SNoLev}}}}~{{z}}}}}}\unicdcdcdhd\\
{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}})}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}}~{{z}}}}).\\

\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9398b7aeb19cac8a975ba6fbd95f5b59201e4420910809555431c85cad9a916b \\
Pure Prop Address: & TMU2SS6hDznnxkCmvyEBsebcgVDL1S1H2x2 \\
Theory Prop Id: & 66456cd00f96a66ea2d8766a84096f51528bb65bd260c0bd0ee0a6d609286810 \\
Theory Prop Address: & TMMddRAHYAMxfbYKpGLowF2aZGsCS8EmE5Q
\end{tabular}
\end{center}
\end{theorem}
\begin{verbatim}
SNoCutP_SNoCut_impred
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfdfoepgdefhehpfjgngahchfgeg} 
$$
\begin{array}{c}
{\unicdcdadad L R .  }{{{{{{{\mname{{SNoCutP}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{o}}}{ .  }\\
({{{{\mname{{SNo}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}}\unicdbdjdcd\\
{{{{\mname{{SNoLev}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}} \in\\
  {{{{\mname{{ordsucc}}}}~{{({{{{({{\unicdcddedd _{{x \in {{L}}}}{{\mname{{ordsucc}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}})}}\unicdcdcdbg{{({{\unicdcddedd _{{y \in {{R}}}}{{\mname{{ordsucc}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}})}}}})}}}}\unicdbdjdcd\\
{{({{\unicdcdadad x \in {{L}} .  {{{{x}}<{{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}}}}}})}}\unicdbdjdcd\\
{{({{\unicdcdadad y \in {{R}} .  {{{{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}}<{{y}}}}}})}}\unicdbdjdcd\\
(\unicdcdadad z .  {{{{\mname{{SNo}}}}~{{z}}}}\unicdbdjdcd
{{({{\unicdcdadad x \in {{L}} .  {{{{x}}<{{z}}}}}})}}\unicdbdjdcd
{{({{\unicdcdadad y \in {{R}} .  {{{{z}}<{{y}}}}}})}}\unicdbdjdcd\\
{{{{{{\mname{{SNoLev}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}} \subseteq {{{{\mname{{SNoLev}}}}~{{z}}}}}}\unicdcdcdhd\\
{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}})}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}}~{{z}}}})\\
\unicdbdjdcd{{p}})\\
{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 02c4ae1ea5a415b6ae779362a18ac7cc3070a6c3c3929db683bbe6328a699c64 \\
Pure Prop Address: & TMZsxeiR2ByApNxNBoTaPP9YjGAmivj7Pn1 \\
Theory Prop Id: & 5bcc8fbe0029984d17a2e449797a375c7e0a555e9ac3699de6cb6126199cfb8e \\
Theory Prop Address: & TMLavVhWLmAEpRZmDjMboZyS2VSTuS99y2H
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCutP_L_0
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfmepfad} ${\unicdcdadad L .  {{({\unicdcdadad x \in {L} .  {{\mname{SNo}}~{x}}})}\unicdbdjdcd{{{\mname{SNoCutP}}~{L}}~{0}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 47329ccaff17b8de852f941e3476b6e2a6155b82363ef200acdb240a53b6fda3 \\
Pure Prop Address: & TMTjdd2ksvbLqB5FsqRdHpicLo8hBbx6oUB \\
Theory Prop Id: & ba61e1cdd15d766dc08200012803abfdba4ca254c084a1b0a956a1cce0e38bec \\
Theory Prop Address: & TMc8bJkCqfKyT4pUBfyifSdGoAoYMJU3VNT
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCutP_0_R
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfadpfcf} ${\unicdcdadad R .  {{({\unicdcdadad x \in {R} .  {{\mname{SNo}}~{x}}})}\unicdbdjdcd{{{\mname{SNoCutP}}~{0}}~{R}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 54eb826f8bd3be24b1ac7e770fe0c82339a00e86e3aa6ad66d6c724dfb7ef61d \\
Pure Prop Address: & TMYBbiGfDG8mCyX4fru38L5jAWs9jdk1E22 \\
Theory Prop Id: & a06bcdb281c727dba6fa27a17f7f8cf4ed415ef56fdd112760f4ab9f181bdd15 \\
Theory Prop Address: & TMVtzjAnWmMCfmCH1umdVeisYihbHtUcePx
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCutP_0_0
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfadpfad} ${{{\mname{SNoCutP}}~{0}}~{0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3b144eeddb1a1e1da010e706f7ae7f7d3118085be6517aa02ae9e9baaabdbf9e \\
Pure Prop Address: & TMFvHEjaKbqPcs4FFJwfso2a1BUbAJ1s8cW \\
Theory Prop Id: & a5144d076fe9899c5c9e19256598fb91665b7f4ccdb1496e795e8e54b2d4dbd1 \\
Theory Prop Address: & TMMq2uhqdU4HWokrwXJLACzASFLaAXQaemQ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCut_0_0
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehpfadpfad} ${{{{\mname{SNoCut}}~{0}}~{0}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6a1f81c2c7f937ee3a0d8974a08070c8b8027c1f662f906b43877504c8d43386 \\
Pure Prop Address: & TMXsrRAvtrifruw52jjYBXPcZSqXpXBYS6i \\
Theory Prop Id: & abbffb386e047e8707315aa435470d2a1b93eb4b65743a2e3d1e70c9bfac9822 \\
Theory Prop Address: & TMNDz4jiqD3F4kCxcNG6hy3DkR89EsDykLk
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNoLt_In
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgmeehpfjeog} ${\unicdcdadad \alpha \beta .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{{\alpha}<{\beta}}\unicdbdjdcd{{\alpha} \in {\beta}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d6f066102af8887431439634f1b57a9b0617aba84ba72851e224f021d81ee1fa \\
Pure Prop Address: & TMboKANLD7F64pb6YHEZed6jmzXMjejG8Hn \\
Theory Prop Id: & de775bfed53c50ad2f8415b8603d1aa62bc9e72e6af350f83a6b199c85e37f40 \\
Theory Prop Address: & TMY34unv2doasrWYaYXyGVaCNDYHke47nRH
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNoLe_Subq
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgmefgpfdffhcgbh} ${\unicdcdadad \alpha \beta .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{{\alpha}\unicdcdgded{\beta}}\unicdbdjdcd{{\alpha} \subseteq {\beta}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c7c906306e5db042974931e626682e6d001f1926ae5d425fb53e074ecbc4a4b6 \\
Pure Prop Address: & TMUky22qrn52gwZasuEjEB8QeZ16vSaUQda \\
Theory Prop Id: & 7447913a7365f17ac2bd36550d83a144dba2081fdb9554f75855b60e9ab366d6 \\
Theory Prop Address: & TMVmvfhvMNFxdUKaz49Zx12oefEK5cmZnmc
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgdfpf}
  We define $\mname{SNoS\_}$ to be
  $${\uniddcgcg \alpha .  {\{x \in {{\mname{Power}}~{({{\mname{SNoElts\_}}~{\alpha}})}}|{\unicdcdaddd \beta \in {\alpha} .  {{{\mname{SNo\_}}~{\beta}}~{x}}}\}}}$$
  of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & d5069aa583583f8fbe5d4de1ba560cc89ea048cae55ea56270ec3dea15e52ca0 \\
Pure Object Address: & TMYeiHkopAzUFpJHiX6DyQTk3GQ6WaqpPTK \\
Theory Object Id: & 4680ceebcb1ec94f9ea6976bf96fcfc7658bb190d9a29c5379cc475cc08324a0 \\
Theory Object Address: & TMS7Rrpy3RhKZhHAm1BhXAbgLBRoK6QSvXq
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
SNoS_E
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdfpffe} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad x \in }{{{{{\mname{{SNoS\_}}}}~{{\alpha}}}}}{ .  }{{\unicdcdaddd \beta \in {{\alpha}} .  {{{{{{\mname{{SNo\_}}}}~{{\beta}}}}~{{x}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 970ce204c4c603af34a26281cf79119bce7edf82baa409d90d59caebd267c095 \\
Pure Prop Address: & TMRdGNRDtsqoEYn5nNq2tqy2tRb7GnvaRGr \\
Theory Prop Id: & bb731c8dff4704c5eeb508b1721d9396d9976279c8d5f0cd55142b3de33acf35 \\
Theory Prop Address: & TMTy8u2pBq8VtTTWH867d9E9g248LG9VRPU
\end{tabular}
\end{center}

\end{theorem}
\section{TaggedSets2}\label{efbghghgfgegdffgehdhcd}Let $tag : {{\iota}\unicdbdjdcd{\iota}}$ be ${\uniddcgcg \alpha .  {{{\mname{SetAdjoin}}~{\alpha}}~{\{{1}\}}}}$.
{\bf{Notation.}} We use $\posthc{-}$ as a postfix operator  corresponding to applying term ${tag}$.
\begin{verbatim}
SNoS_I
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdfpfje} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad x .  }{\unicdcdadad \beta \in }{{{\alpha}}}{ .  }{{{{{{{\mname{{SNo\_}}}}~{{\beta}}}}~{{x}}}}}{\unicdbdjdcd}{{{{x}} \in {{{{\mname{{SNoS\_}}}}~{{\alpha}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f5227dbc4fc6188f3716bce1eb047d891fe917014fdf489301acb45e2c964b31 \\
Pure Prop Address: & TMMNh5GJRjfU6R9jYR9oSB51a82UR8JuQEx \\
Theory Prop Id: & 04f24bbca0962b615517b9ad07d87ff686ee791137b3963827817df34a43adf4 \\
Theory Prop Address: & TMKCtpebj2a31twyQ52NMFQ1XPqjxuuqVSL
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoS_I2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdfpfjecd} 
$$
\begin{array}{c}
{\unicdcdadad x y .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoLev}}}}~{{x}}}} \in {{{{\mname{{SNoLev}}}}~{{y}}}}}}}{\unicdbdjdcd}{{{{x}} \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7a14c7d2ef8827bceb18bb81afdc45412f5d9fea9a904ca562bfb0f81d89d276 \\
Pure Prop Address: & TMJ7HAvGksViV5vVRG5CZDkJu7DwFRhbuix \\
Theory Prop Id: & b4042d0ea055be8fc3da99d768f853e3498c984596760bcc2ea38e088f32c744 \\
Theory Prop Address: & TMKRKaYnYhhkyGaZmUUzgdvAQzZwXTCobPz
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoS_Subq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdfpfdffhcgbh} 
$$
\begin{array}{c}
{\unicdcdadad \alpha \beta .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{\mname{{ordinal}}}}~{{\beta}}}}}{\unicdbdjdcd}{{{{{\alpha}} \subseteq {{\beta}}}}}{\unicdbdjdcd}{{{{{{\mname{{SNoS\_}}}}~{{\alpha}}}} \subseteq {{{{\mname{{SNoS\_}}}}~{{\beta}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6828e7536cfb2fa9917d83ea810ef035d006db4740719d8a39bb37385dbd855d \\
Pure Prop Address: & TMUi6zuABJzGjt2kXYnMDgkM42sqaRJMMSb \\
Theory Prop Id: & d928b8db0b481798073f21b6dda22ac8c152f53325fda84e77c9b8ec06d1d6e3 \\
Theory Prop Address: & TMcJgHjSXytYqLvRxz9hc3gMG8529moaiYm
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_uniq2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpffhogjgbhcd} ${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad x .  {{{{\mname{SNo\_}}~{\alpha}}~{x}}\unicdbdjdcd{{{\mname{SNoLev}}~{x}}={\alpha}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fbb46148e53cbe3acaacc8b0b0d7126ebd90194aeb82198e6da28510ce8036d1 \\
Pure Prop Address: & TMPvNq7g9BwshVJwKXtCkaoFEnCVHUogHas \\
Theory Prop Id: & f48d47fd7fa9af8472cc826916e9dcead2eaad0845f212108c42b4b8599893ef \\
Theory Prop Address: & TMUESvsETvw3i1PT8zcwLiWdQvbBBUe1Xw5
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoS_E2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdfpffecd} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad x \in }{{{{{\mname{{SNoS\_}}}}~{{\alpha}}}}}{ .  }{\unicdcdadad p : }{{{o}}}{ .  }\\
{{{({{{{{{{{\mname{{SNoLev}}}}~{{x}}}} \in {{\alpha}}}}\unicdbdjdcd{{{{{{\mname{{ordinal}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}\unicdbdjdcd{{{{{{\mname{{SNo}}}}~{{x}}}}\unicdbdjdcd{{{{{{{{\mname{{SNo\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}~{{x}}}}\unicdbdjdcd{{p}}}}}}}}}})}}}\\
{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & deb03168c1f55ab1d7d74277ebc4f8f33c87b28b16265d557f144f3a25032de8 \\
Pure Prop Address: & TMRabQVdMnYRsChGTYCN42TFfnEWrGyAXz2 \\
Theory Prop Id: & 62b490ef867368eaf329040a83310276c2ad2d046f5274d8c507fe860ea3918f \\
Theory Prop Address: & TMb5fbntT2N1CnACiMUeKM3JYGV5og2W5ZC
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoS_In_neq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdfpfjeogpfogfgbh} ${\unicdcdadad w .  {{{\mname{SNo}}~{w}}\unicdbdjdcd{\unicdcdadad x \in {{\mname{SNoS\_}}~{({{\mname{SNoLev}}~{w}})}} .  {{x}\unicdcdgdad{w}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ab85fd9a0fdd5f7f96ca2f75d5fd279bf5e0232396c952796f3fc30b522d427d \\
Pure Prop Address: & TMFyX2RhEpYBCTYG6ZNQWUrYk2pQxTkZujK \\
Theory Prop Id: & d5dbd7ce116b1b3b16cbebfad9387d99e477865cfc62e88ffd6e229c2248d86f \\
Theory Prop Address: & TMZeUEJeQ6MejSErRQj2VVEk9N7K2zaE9uV
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoS_SNoLev
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdfpfdfoepgmefggh} 
$$
\begin{array}{c}
{\unicdcdadad z .  }{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{{{{z}} \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{ordsucc}}}}~{{({{{{\mname{{SNoLev}}}}~{{z}}}})}}}})}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a9ba1fc0964d1c60416e8e5bf12e8514139434ea6fe325257b505b527b64eee1 \\
Pure Prop Address: & TMW8wR8JRK9MYLigu7413p9VCMwfzGT34AQ \\
Theory Prop Id: & 8efb88111c118260d0d7ef1b5cdc5079c9b4a3be002c1cb750eb87a510583725 \\
Theory Prop Address: & TMbmaGPAkU5fqbmKhk6PvZkArv7GqJuJRmY
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgme}
We define $\mname{SNoL}$ to be ${\uniddcgcg z .  {\{x \in {{\mname{SNoS\_}}~{({{\mname{SNoLev}}~{z}})}}|{{x}<{z}}\}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 8cd812b65c6c466abea8433d21a39d35b8d8427ee973f9bb93567a1402705384 \\
Pure Object Address: & TMUvMXBQ19xuZkaN1Qr1oL9x2RgVQBhPiEP \\
Theory Object Id: & 48bcf841807be9a2a8a7fdaa4062b7f700ed225f5219ffd13f26782d239d11ae \\
Theory Object Address: & TMGV9huV8KZCMkFQWMGc7fqW2LnTPhHVw9y
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:dfoepgcf}
We define $\mname{SNoR}$ to be ${\uniddcgcg z .  {\{y \in {{\mname{SNoS\_}}~{({{\mname{SNoLev}}~{z}})}}|{{z}<{y}}\}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 73b2b912e42098857a1a0d2fc878581a3c1dcc1ca3769935edb92613cf441876 \\
Pure Object Address: & TMSvKHoWGSuy2M3PmTt5s6iktHABF7qYPcr \\
Theory Object Id: & 0cfe870d225f8cba537d2fe6552d285b3601e6ba81a7b2c47d955f4d490e3aba \\
Theory Object Address: & TMH8PQYnqVae2kcUxoh7NLmWhd5qkNUZU8E
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
SNoCutP_SNoL_SNoR
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfdfoepgmepfdfoepgcf} ${\unicdcdadad z .  {{{\mname{SNo}}~{z}}\unicdbdjdcd{{{\mname{SNoCutP}}~{({{\mname{SNoL}}~{z}})}}~{({{\mname{SNoR}}~{z}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 52486657c66c30f3a313a97245051168bef3ab087bda6a6d0afdc7a3a1f8843e \\
Pure Prop Address: & TMQrzjSnfMHhEaKZxuYJx3zZLL9WfuFojym \\
Theory Prop Id: & 7aba571de6f479a36c61b1efbc3a3061f907bdbc5585aca815e7b788f51bd0a2 \\
Theory Prop Address: & TMVyVwNCUniGutAsBRT2zh5MKBDjX715Vgi
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoL_E
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmepffe} 
$$
\begin{array}{c}
  {\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{\unicdcdadad w \in }{{{{{\mname{{SNoL}}}}~{{x}}}}}{ .  }{\unicdcdadad p : }{{{o}}}{ .  }\\
  {({{{{{{\mname{{SNo}}}}~{{w}}}}\unicdbdjdcd{{{{{{{{\mname{{SNoLev}}}}~{{w}}}} \in {{{{\mname{{SNoLev}}}}~{{x}}}}}}\unicdbdjdcd{{{{{{w}}<{{x}}}}\unicdbdjdcd{{p}}}}}}}})}\\
{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & cead4f50e23f6816dc9703016f0ed79b8bdbc32a7c09a0f9af78552aeebba9c0 \\
Pure Prop Address: & TMZmXPiB4BB3YvdBwZTD97bGggD7rSLF6eD \\
Theory Prop Id: & 7ccd8b9dc0186ef905a8d93999fdc69eb492737ff71320a53317c135b5d63fb4 \\
Theory Prop Address: & TMYKcvufFZ2FLebZaXDWknwUKjkcdzGsQTZ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoR_E
\end{verbatim}
\begin{theorem}\label{thm:dfoepgcfpffe} 
$$
\begin{array}{c}
  {\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{\unicdcdadad z \in }{{{{{\mname{{SNoR}}}}~{{x}}}}}{ .  }{\unicdcdadad p : }{{{o}}}{ .  }\\
  {{{({{{{{{\mname{{SNo}}}}~{{z}}}}\unicdbdjdcd{{{{{{{{\mname{{SNoLev}}}}~{{z}}}} \in {{{{\mname{{SNoLev}}}}~{{x}}}}}}\unicdbdjdcd{{{{{{x}}<{{z}}}}\unicdbdjdcd{{p}}}}}}}})}}}\\
{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & efca05b2e45ac1d5303c49c52a3636406f36f5a77d8812e0200eaefc9712df2c \\
Pure Prop Address: & TMbsBxRcWxxu6mkgykGDi4yRnEUFXwJJe4Z \\
Theory Prop Id: & 7dca045bf869829b43204d36be8d7632a5061acf12e145a99d1cd8d9be1edfda \\
Theory Prop Address: & TMUsPcK3wqDZwDie4ZMQqUBLgLnuT57PPnY
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoL_SNoS
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmepfdfoepgdf} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{\unicdcdadad w \in }{{{{{\mname{{SNoL}}}}~{{x}}}}}{ .  }{{{{w}} \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6e3a2292c3997c29e78fcb520858907ddc605dae52383648872fd6c7ce45f0c1 \\
Pure Prop Address: & TMLcRAnSRqhUGWvLLTk89wzM2sZ4wvddz98 \\
Theory Prop Id: & ded78aa4dc105de48885dd5e055230505548286bce7f04f3f5050316c2c04cd0 \\
Theory Prop Address: & TMPHFqjx1BZx94gsexg3cyERVChRcfu9aMh
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoR_SNoS
\end{verbatim}
\begin{theorem}\label{thm:dfoepgcfpfdfoepgdf} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{\unicdcdadad z \in }{{{{{\mname{{SNoR}}}}~{{x}}}}}{ .  }{{{{z}} \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 98051c05991e35d56daa447cc7976259612d6048fa82f9a44da11c916b906848 \\
Pure Prop Address: & TMJ2p8noSCp43KaJq9tZFaAKHrVfWm26tto \\
Theory Prop Id: & e13f497f5dab3650206c4362046a1ec477ed8e40d771a06657c3db32f3b247b0 \\
Theory Prop Address: & TMJz37dSJBFjqARSMYVzEuc7L1KYXuTUBFc
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoL_SNoS_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmepfdfoepgdfpf} ${\unicdcdadad z .  {{{\mname{SNoL}}~{z}} \subseteq {{\mname{SNoS\_}}~{({{\mname{SNoLev}}~{z}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 73f5b2cc52ef1daf3b3493d4b3a1b56e0f6a7fedf019873cb56ed5165fc129f6 \\
Pure Prop Address: & TMVShTkL3TuhDkq8iCTfj9xfEecNaq6hZTr \\
Theory Prop Id: & 409713becd3f086d3979ea36c2db06311233c54602ab8b2acc36bc0e856661f0 \\
Theory Prop Address: & TMacyLTfdfpQiEh9iWRH45thaEffs457QES
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoR_SNoS_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgcfpfdfoepgdfpf} ${\unicdcdadad z .  {{{\mname{SNoR}}~{z}} \subseteq {{\mname{SNoS\_}}~{({{\mname{SNoLev}}~{z}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 16bdfc8b4f768a6b37a41115958b385c5a1ec67f13570e242fd1933e158b499f \\
Pure Prop Address: & TMSz3ALjKBot9ZKASXvvTfDJErg7bfASa2J \\
Theory Prop Id: & 2fe1c08fc3923f123fffe6c7d8be4b5955d977da8e53d13d4d6c6f092bdcc16d \\
Theory Prop Address: & TMbbztsQAvihxVwkJ9kvqAoNEe7VRXcgYMR
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoL_I
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmepfje} 
$$
\begin{array}{c}
{\unicdcdadad z .  }{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoLev}}}}~{{x}}}} \in {{{{\mname{{SNoLev}}}}~{{z}}}}}}}{\unicdbdjdcd}{{{{{x}}<{{z}}}}}{\unicdbdjdcd}{{{{x}} \in {{{{\mname{{SNoL}}}}~{{z}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ddd91ba46f8cb49763a2e555070ff9cf0d25b89336d5efd26582ff2019e6e033 \\
Pure Prop Address: & TMFWNR9kjhSxmC1PpWAxELFPMCQrQDgS3ii \\
Theory Prop Id: & 4f3d049c65e44bceaa7abebe992ac83d5e90855d0e811e33cecee9e4c04e2717 \\
Theory Prop Address: & TMV173uXJoeNM1as5esbiKuaj6zEgDhUJka
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoR_I
\end{verbatim}
\begin{theorem}\label{thm:dfoepgcfpfje} 
$$
\begin{array}{c}
{\unicdcdadad z .  }{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{\unicdcdadad y .  }{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoLev}}}}~{{y}}}} \in {{{{\mname{{SNoLev}}}}~{{z}}}}}}}{\unicdbdjdcd}{{{{{z}}<{{y}}}}}{\unicdbdjdcd}{{{{y}} \in {{{{\mname{{SNoR}}}}~{{z}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 97693c3a27a8f00a3262b80f853884f0e0d68e482cbff49cb642b3f14d256b71 \\
Pure Prop Address: & TMTPFvahbCyh9yFHZ4zMiRoweJ7ZXMnfLx8 \\
Theory Prop Id: & 383e88e28a8ea102937169b00da3f18efb2523746b17c0fc524867476a449c5a \\
Theory Prop Address: & TMSrHC17ffyA2enfcx5o1B7iUJn1JV3jwSc
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_eta
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgehbg} ${\unicdcdadad z .  {{{\mname{SNo}}~{z}}\unicdbdjdcd{{z}={{{\mname{SNoCut}}~{({{\mname{SNoL}}~{z}})}}~{({{\mname{SNoR}}~{z}})}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3cc369ecf11e43c8a7a6cd9176f7edf2e2d6f77128e0bb8b46dd7e9adb3afd18 \\
Pure Prop Address: & TMMbZPXPQWq2cEQg1eagYj6NwweZYGCHjiJ \\
Theory Prop Id: & dc9273d2d30d56ac9a4eff2b975bce32ce2e1e109d620f9318846e4176dda05b \\
Theory Prop Address: & TMZQ7w25dF2ftfDkjDshscxccT898iSijZ7
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCutP_SNo_SNoCut
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfdfoepgpfdfoepgdefheh} ${\unicdcdadad L R .  {{{{\mname{SNoCutP}}~{L}}~{R}}\unicdbdjdcd{{\mname{SNo}}~{({{{\mname{SNoCut}}~{L}}~{R}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9621aa2a6018b3eb1183066003e6daeb26c90761edfa02dd7e872bf831ccb4f8 \\
Pure Prop Address: & TMNKaoM4RaxsmapryqxboP9xADcYUVy5VVL \\
Theory Prop Id: & ae46e89c23c7cd0b1fbccb0a1e97f383308cec395b5c73ec3d890f6d1641f001 \\
Theory Prop Address: & TMMkfWp7u1CPUEtc9sUMWAX6ZkHiFKL3Tsx
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCutP_SNoCut_L
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfdfoepgdefhehpfme} ${\unicdcdadad L R .  {{{{\mname{SNoCutP}}~{L}}~{R}}\unicdbdjdcd{\unicdcdadad x \in {L} .  {{x}<{{{\mname{SNoCut}}~{L}}~{R}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 984a30c42556263d781f855feaa7c03ffd5961f0b9c95159ea8302b3ca16467c \\
Pure Prop Address: & TMKfLyuDzXLNY7PtnVTeZe7eLvSaxAynvFi \\
Theory Prop Id: & 4bca14807188033563b691a30256c970867594d77736618ddaeed7f6ead01b56 \\
Theory Prop Address: & TMcuZs8Sau5dRwSYifuF8HCBdKnRRmSKZYt
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCutP_SNoCut_R
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfdfoepgdefhehpfcf} ${\unicdcdadad L R .  {{{{\mname{SNoCutP}}~{L}}~{R}}\unicdbdjdcd{\unicdcdadad y \in {R} .  {{{{\mname{SNoCut}}~{L}}~{R}}<{y}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9161db8c6978c922fbaa8ef29650182e4dc42c4f7e4bbe22e34280baae014e7e \\
Pure Prop Address: & TMdihdbyAcDNSyUkD3TdUvmSMiU3VY6TxqA \\
Theory Prop Id: & 7c3d12a89f726793a0a60bd44097c71eaee25e353ad706ea323bc543dbf26c62 \\
Theory Prop Address: & TMPBd6BZmniMTRdKnM7fbpYBkR7PiBbKsPm
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCutP_SNoCut_fst
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehafpfdfoepgdefhehpfggdheh} 
$$
\begin{array}{c}
{\unicdcdadad L R .  }{{{{{{{\mname{{SNoCutP}}}}~{{L}}}}~{{R}}}}}{\unicdbdjdcd}{\unicdcdadad z .  }{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{{{({{\unicdcdadad x \in {{L}} .  {{{{x}}<{{z}}}}}})}}}{\unicdbdjdcd}{{{({{\unicdcdadad y \in {{R}} .  {{{{z}}<{{y}}}}}})}}}{\unicdbdjdcd}\\
{{{{{{\mname{{SNoLev}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}} \subseteq {{{{\mname{{SNoLev}}}}~{{z}}}}}}\unicdcdcdhd\\
{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}})}}}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}}~{{z}}}}.\\
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b52eb75a2c789c7e4e20b3f5065952940fdfe39af85380d08e4788785181a8fa \\
Pure Prop Address: & TMGpKvy9aLQxgW1tTVWtNezptYGvyfjG5Lj \\
Theory Prop Id: & 8362f8acd554e760437ebf6fcb6ddf58a0b184474b06a457ee1b26d03f2fb41a \\
Theory Prop Address: & TMKQruVQCojmwaMqYHkmFCxgxgAAJu5hamZ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCut_Le
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehpfmefg} 
$$
\begin{array}{c}
{\unicdcdadad L1 R1 L2 R2 .  }{{{{{{{\mname{{SNoCutP}}}}~{{L1}}}}~{{R1}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoCutP}}}}~{{L2}}}}~{{R2}}}}}{\unicdbdjdcd}\\
{{{({{\unicdcdadad w \in {{L1}} .  {{{{w}}<{{{{{{\mname{{SNoCut}}}}~{{L2}}}}~{{R2}}}}}}}})}}}{\unicdbdjdcd}{{{({{\unicdcdadad z \in {{R2}} .  {{{{{{{{\mname{{SNoCut}}}}~{{L1}}}}~{{R1}}}}<{{z}}}}}})}}}{\unicdbdjdcd}\\
{{{{{{{{\mname{{SNoCut}}}}~{{L1}}}}~{{R1}}}}\unicdcdgded{{{{{{\mname{{SNoCut}}}}~{{L2}}}}~{{R2}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & bb8cdcd2d5fd2f00037512dfe3c9dc90186e99a11943fa34839c21c6d21327f9 \\
Pure Prop Address: & TMNRFtNxDRZszFXoDioSFyUkctJH577Wgtm \\
Theory Prop Id: & 4237313994e58f38bf2a6019231c10978a504fc59d522337bd78a558d606fc9d \\
Theory Prop Address: & TMU9RWn6464F25Pra7jPLPqiPfwa8i21YQ3
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoCut_ext
\end{verbatim}
\begin{theorem}\label{thm:dfoepgdefhehpffgiheh} 
$$
\begin{array}{c}
{\unicdcdadad L1 R1 L2 R2 .  }{{{{{{{\mname{{SNoCutP}}}}~{{L1}}}}~{{R1}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoCutP}}}}~{{L2}}}}~{{R2}}}}}{\unicdbdjdcd}\\
{{{({{\unicdcdadad w \in {{L1}} .  {{{{w}}<{{{{{{\mname{{SNoCut}}}}~{{L2}}}}~{{R2}}}}}}}})}}}{\unicdbdjdcd}{{{({{\unicdcdadad z \in {{R1}} .  {{{{{{{{\mname{{SNoCut}}}}~{{L2}}}}~{{R2}}}}<{{z}}}}}})}}}{\unicdbdjdcd}\\
{{{({{\unicdcdadad w \in {{L2}} .  {{{{w}}<{{{{{{\mname{{SNoCut}}}}~{{L1}}}}~{{R1}}}}}}}})}}}{\unicdbdjdcd}{{{({{\unicdcdadad z \in {{R2}} .  {{{{{{{{\mname{{SNoCut}}}}~{{L1}}}}~{{R1}}}}<{{z}}}}}})}}}{\unicdbdjdcd}\\
{{{{{{{{\mname{{SNoCut}}}}~{{L1}}}}~{{R1}}}}={{{{{{\mname{{SNoCut}}}}~{{L2}}}}~{{R2}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c1078eaf6d9d767331c2adfc239cf6d058aa0386b0e05639ec15f00d6bc8651b \\
Pure Prop Address: & TMQEDzfjypV741w6WAoTJECPB5vgdYS7NXj \\
Theory Prop Id: & f66618e40670b959c354add07597e52634943b95783e7550a46245c83744c351 \\
Theory Prop Address: & TMQc8BkWk3YDRQ3ZDJXw13dYU48b624zt8h
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLt_SNoL_or_SNoR_impred
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehpfdfoepgmepfpgchpfdfoepgcfpfjgngahchfgeg} 
$$
\begin{array}{c}
{\unicdcdadad x y .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{x}}<{{y}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{o}}}{ .  }{{{({{\unicdcdadad z \in {{{{\mname{{SNoL}}}}~{{y}}}} .  {{{{{{z}} \in {{{{\mname{{SNoR}}}}~{{x}}}}}}\unicdbdjdcd{{p}}}}}})}}}{\unicdbdjdcd}\\
{{{({{{{{{x}} \in {{{{\mname{{SNoL}}}}~{{y}}}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{{{({{{{{{y}} \in {{{{\mname{{SNoR}}}}~{{x}}}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f3b15a8dbd76f9e501e20410a95606c6e4d8a53c81914a627374d46d3d907b2f \\
Pure Prop Address: & TMRDwmSiYvVpLRjvbAK1MWfXAVsfvotprEu \\
Theory Prop Id: & ffc0bf982d5e8f4c56f088c7cc742ee4ab02e0c490f0c7638cbc614dc1677aca \\
Theory Prop Address: & TMcckGGW8ynGG29S7CCB8qwfJasDqZZoUqS
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoL_or_SNoR_impred
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmepfpgchpfdfoepgcfpfjgngahchfgeg} 
$$
\begin{array}{c}
{\unicdcdadad x y .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{o}}}{ .  }{{{({{{{{{x}}={{y}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{{{({{\unicdcdadad z \in {{{{\mname{{SNoL}}}}~{{y}}}} .  {{{{{{z}} \in {{{{\mname{{SNoR}}}}~{{x}}}}}}\unicdbdjdcd{{p}}}}}})}}}{\unicdbdjdcd}\\
{{{({{{{{{x}} \in {{{{\mname{{SNoL}}}}~{{y}}}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{{{({{{{{{y}} \in {{{{\mname{{SNoR}}}}~{{x}}}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{{{({{\unicdcdadad z \in {{{{\mname{{SNoR}}}}~{{y}}}} .  {{{{{{z}} \in {{{{\mname{{SNoL}}}}~{{x}}}}}}\unicdbdjdcd{{p}}}}}})}}}{\unicdbdjdcd}\\
{{{({{{{{{x}} \in {{{{\mname{{SNoR}}}}~{{y}}}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{{{({{{{{{y}} \in {{{{\mname{{SNoL}}}}~{{x}}}}}}\unicdbdjdcd{{p}}}})}}}{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f8a5e25fc885d4cd85a70c63657ca7547fbf75b02f27b82464c042547cc677f6 \\
Pure Prop Address: & TMKsCmKUpLx2BhcK6jEDUspKVKvE5k5QiRa \\
Theory Prop Id: & 68ba9809e987e498558a641b25900231b82622d983653583a201a59a4d1c0b55 \\
Theory Prop Address: & TMKWiFqjuwfNEiTTVN1gfvvjWVrJhQj1gqg
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNo_
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgpf} ${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{SNo\_}}~{\alpha}}~{\alpha}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 4731fd4a3d0693e855e835e22a21f5e5977b26765eba190ec58fe9daa4d2afe8 \\
Pure Prop Address: & TMWHKTo9dcgkBtGz7uaXaYTj4kgHyCDDyAP \\
Theory Prop Id: & 7754e54755e12ec8adcd2c120c5d5474ec249b91da929ca1e7981dd9d443b092 \\
Theory Prop Address: & TMbYC7hUX1cj11jwA8NrmcFtWzGyRNNcP3L
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNoL
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgme} ${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{SNoL}}~{\alpha}}={{\mname{SNoS\_}}~{\alpha}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fef60879d9dcebd791ff77a26af98cae1b05821d9fe106e7d892b781e23e857d \\
Pure Prop Address: & TMKCDKt4HyT8bYK3wPMhnwhhHEbQ5xacdeK \\
Theory Prop Id: & 95823e3d326f1172e3884b6aa394d6e9d59d2ddf7578bca2e2f2aa935524fd70 \\
Theory Prop Address: & TMd2zwjPUXZL6Vy46VFW4gwpc2zig21PKRe
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNoR
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgcf} ${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{SNoR}}~{\alpha}}={\mname{Empty}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 07c69afcba017b5ee2b34caa8d7b084b72822e15942c9a589e012fc3354dadc3 \\
Pure Prop Address: & TMEszERJsS849AqzpPGdTFGPRhenRZSY14a \\
Theory Prop Id: & f853d9f1e9d0ea7f8c0d0f6837e18502503c2b8c0a0f017d5f07a8be9f70fda3 \\
Theory Prop Address: & TMRXLgDuMtvBV5siWdFYrySgsWpvnxN9925
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNoCutP
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgdefhehaf} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{{{\mname{{SNoCutP}}}}~{{({{{{\mname{{SNoS\_}}}}~{{\alpha}}}})}}}}~{{\mname{{Empty}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 1e4d0d44a7461a68881734e0e91b185e38404b2e73f6e59f99b11fe839407b3d \\
Pure Prop Address: & TMW1UvmeHxUQLv7BE98ap334SUfCm8Fu73Y \\
Theory Prop Id: & 0cfe8be75650c6e5a146099344f74fb66757e93b2cd2edac1ae4deefc984dc64 \\
Theory Prop Address: & TMRkZRJr4waasoDGCAVbUFmg6pCVbwcgmgu
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNoCut_eta
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgdefhehpffgehbg} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{{{{\alpha}}={{{{{{\mname{{SNoCut}}}}~{{({{{{\mname{{SNoS\_}}}}~{{\alpha}}}})}}}}~{{\mname{{Empty}}}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & a115354a2357cc66a492e6c27636f873c89ca659eeae89203adba376b623519f \\
Pure Prop Address: & TMSVcnu3qbhNj765CNa5JVutvQs8ZAUZbac \\
Theory Prop Id: & 1f113a2b452bb89a7755a221d966a72de174752e7ff3e1d939ad96691e455817 \\
Theory Prop Address: & TMFKp4mHaoLW3mEp9qD9XgwpSzYRMvq34F2
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_0
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfad} ${{\mname{SNo}}~{0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c8204b8b1882d371baace062a2960d93b16000698c7d6a09a142db659b3f1fca \\
Pure Prop Address: & TMN5B5ciEPfW7oQwW9YQcm3vLvkEtUZSZnt \\
Theory Prop Id: & 94a310917df9f628979dfd6d1313a09153628ffb779d20e4011addf6dc4f2d0d \\
Theory Prop Address: & TMFnaFB9Pt5pY6PgwroCRcRZDgrkfSYBuSa
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_0
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpfad} ${{{\mname{SNoLev}}~{0}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 45a0f7a396d9f9d63598766e2614400055684c37fc87764bd32b3d694a4f878e \\
Pure Prop Address: & TMJQ1Ke47ELvhoXkPaZRPfgMk7d8RGCakfG \\
Theory Prop Id: & 3527a4181b9135af63b4982294e24684d20c27740e7c8367f29e25ad12624fa4 \\
Theory Prop Address: & TMLKkLmPfzCPj82cSv1tmqe99GdjnMnFnvv
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoL_0
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmepfad} ${{{\mname{SNoL}}~{0}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c01c2ba737a470d613bec196a861574fc97eac3285aae68253f8ce0a1199ed06 \\
Pure Prop Address: & TMPzAPKfbgig2tHnBj3tDTW5gQHRQr85Y6m \\
Theory Prop Id: & fdaa694c1dad3b51990c33828083df7857c7ce848cddbb249b15569df8cf4b8c \\
Theory Prop Address: & TMVPonV6zmxAnnnAJAxaJ4wWH75boedRK8V
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoR_0
\end{verbatim}
\begin{theorem}\label{thm:dfoepgcfpfad} ${{{\mname{SNoR}}~{0}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7329c7a0141d5faa6d6c735a4606f98929d053ab782ccfee0d527914d8d0bb82 \\
Pure Prop Address: & TMHH23zks5tKNiFan71Zdnpm67D36aSbUoZ \\
Theory Prop Id: & 51075148dc5c48d42db4d40072bca52ac7fb82c5a788b39d6f6b7e7c69cd1ecc \\
Theory Prop Address: & TMajtxBFtVd1rrqznKQdB7sqrYQ1JGnSYS3
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoL_1
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmepfbd} ${{{\mname{SNoL}}~{1}}={1}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 84d688b29c78bb8b93ed259d6f9f639fde3ee97c6f1196e2a383bf07c901a937 \\
Pure Prop Address: & TMSoMwfGW9Mu39jcotcU6thi6J4LWYEx7bn \\
Theory Prop Id: & 15aa793de666a844e874c1d586a19989e272dfba20c5493bdc382a6d38d3b874 \\
Theory Prop Address: & TMTBX6BysYj8xm3zeyBXfyKjENW9x7yzMib
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoR_1
\end{verbatim}
\begin{theorem}\label{thm:dfoepgcfpfbd} ${{{\mname{SNoR}}~{1}}={0}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e6a83a40947575bc277f1b8c0862189045f237a6889acb55f060da536e5d0e57 \\
Pure Prop Address: & TMPLYyxJkuB1KiJuFytWk8RBxXzyup6DPfW \\
Theory Prop Id: & 8da427f5c15d8ec84a8b68f3c128839fd941641f566c3d76a8fc06701af24cd2 \\
Theory Prop Address: & TMVQas5fNZ38HF4hftusV6FZz3JLDZMUN9n
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_max_SNoLev
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfngbgihpfdfoepgmefggh} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{({{\unicdcdadad y \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}} .  {{{{y}}<{{x}}}}}})}}}{\unicdbdjdcd}{{{{{{\mname{{SNoLev}}}}~{{x}}}}={{x}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ff019e872fd598ef8a2a3be672f6bcc7cc7017530d8e62908bad15bf571b16b0 \\
Pure Prop Address: & TMTDt7E8q54KkcysEpcH7qcMKm5WwHpsKqT \\
Theory Prop Id: & 3be24662ae052242ba894f8ccca93140cca84c843c9c37beb7a1560477952006 \\
Theory Prop Address: & TMaz8UPXxX1bFVUXeXSuQwyrGjmQjY46y6C
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_max_ordinal
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfngbgihpfpgchegjgogbgmg} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{({{\unicdcdadad y \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}} .  {{{{y}}<{{x}}}}}})}}}{\unicdbdjdcd}{{{{\mname{{ordinal}}}}~{{x}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 0b230ced38578b81f36ac519fa85acee92a243a563fecccc69ef13c26cc93ac1 \\
Pure Prop Address: & TMJBDQRmWeDvJspje8G3NTmF5iRA5QFjvTX \\
Theory Prop Id: & f28b9a03523768b82a0c58f1bbd347f928da765c1382c04c5d8307d059157531 \\
Theory Prop Address: & TMTYDjsbqqY8KJTxNg5X66HtVW1bn6sTmVP
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:dfoepgpffgihehfgogegad}
We define $\mname{SNo\_extend0}$ to be $${\uniddcgcg x .  {{{\mname{PSNo}}~{({{\mname{ordsucc}}~{({{\mname{SNoLev}}~{x}})}})}}~{({\uniddcgcg \delta .  {{{\delta} \in {x}}\unicdcdcdhd{{\delta}\unicdcdgdad{{\mname{SNoLev}}~{x}}}}})}}}$$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 997d9126455d5de46368f27620eca2e5ad3b0f0ecdffdc7703aa4aafb77eafb6 \\
Pure Object Address: & TMQChUQBGYBvAJNKavV49dHDgfu4qGjxHQV \\
Theory Object Id: & ff4161d4bf42181c2a7fa1c1b810f9d4878846fc9af335cc282ffd21f440f9f4 \\
Theory Object Address: & TMbS5DXTFrYci5baCfTAfaQPuyL4WdRc1rC
\end{tabular}
\end{center}

\end{definition}
\begin{definition}\label{def:dfoepgpffgihehfgogegbd}
We define $\mname{SNo\_extend1}$ to be $${\uniddcgcg x .  {{{\mname{PSNo}}~{({{\mname{ordsucc}}~{({{\mname{SNoLev}}~{x}})}})}}~{({\uniddcgcg \delta .  {{{\delta} \in {x}}\unicdcdcdid{{\delta}={{\mname{SNoLev}}~{x}}}}})}}}$$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 464e47790f44cd38285279f563a5a918d73224c78a9ef17ae1a46e62a95b2a41 \\
Pure Object Address: & TMKMddAcJS6FL2G9uiJereyb1tjokjtKa8Q \\
Theory Object Id: & 8c551b76c450b237e1236b32a569a43ca6b32a062d1d2a013da3dc303569fa68 \\
Theory Object Address: & TMQVotiw6nRDP1Q877RTRTfvRtH76B8baK8
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
SNo_extend0_SNo_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegadpfdfoepgpf} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{\mname{{SNo\_}}}}~{{({{{{\mname{{ordsucc}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}})}}}}~{{({{{{\mname{{SNo\_extend0}}}}~{{x}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d41d3ef86775d62e2fc9b3c9a272033d4ea2944b3c8af763707dfdef923c968f \\
Pure Prop Address: & TMQd3GQMR6hK1kRRtEeaAkFX4ThA8PYbx4s \\
Theory Prop Id: & c8f8ea39e4cdce2253d47eb0646052513914897eaed4338727a5cd163df2b423 \\
Theory Prop Address: & TMH2yeAxqLFK5UHLb9b8dQ6DAmWA6XAxYHL
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend1_SNo_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegbdpfdfoepgpf} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{\mname{{SNo\_}}}}~{{({{{{\mname{{ordsucc}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}})}}}}~{{({{{{\mname{{SNo\_extend1}}}}~{{x}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b7b8669f74b82f6386406947ed40fe7ac42ab269bbd6ea1fa28e5c6678ff4ef4 \\
Pure Prop Address: & TMZZkw7fkUGWbN4TRGNYZKePU3uacXEAcNy \\
Theory Prop Id: & 73bbf256bdb458b2821e16079f5ac4e447c94d365812f86f5c93f8bc91c0cf93 \\
Theory Prop Address: & TMXbECAsrsXbVHqfqLcWLMGLrA5ADZz1k4Y
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend0_SNo
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegadpfdfoepg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{\mname{SNo}}~{({{\mname{SNo\_extend0}}~{x}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & b353a2b441224520a178acfb9f0f817ca34603e17848371ce6ed746729e2d8a2 \\
Pure Prop Address: & TMWGxYp3JwTkkqSrsYrfogqPN1vb2Xorvh1 \\
Theory Prop Id: & 76562ca12828cc8acac6c03cb0db1ff686b05c1fa8f57098d309a8fb90aee94c \\
Theory Prop Address: & TMQtywDaDPXseniJkxkpxbNAdKd38UE2dQ8
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend1_SNo
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegbdpfdfoepg} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{\mname{SNo}}~{({{\mname{SNo\_extend1}}~{x}})}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fa323a2025df47e7721db069b72ed5eee27d1ee3571a8a8af9723ad120927c88 \\
Pure Prop Address: & TMSdMxvikAWSF7zWCcgb3FfvRNkDDipwxVq \\
Theory Prop Id: & 5345a767fd64365f567ddd0c36e71e4585b5269ee70acf1e49fbec29ea6b20f8 \\
Theory Prop Address: & TMY25eBqvG7PbCF867BfH8wXMFyJ3vZVa3z
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend0_SNoLev
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegadpfdfoepgmefggh} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{\mname{{SNoLev}}}}~{{({{{{\mname{{SNo\_extend0}}}}~{{x}}}})}}}}={{{{\mname{{ordsucc}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 26efcd107d12caee8e8ee10adebe189fb528ee43376423b43537d0d980b96057 \\
Pure Prop Address: & TMLE3zBjSiXJKVByLuNppcpDaWc5KxbJsij \\
Theory Prop Id: & 5aa0afa406447583a2ffcc856c3c2270249851120d173ff811fd161bbb5476f1 \\
Theory Prop Address: & TMFDXAViZJBzqyvBjZTxryG6TwYcAsVpsez
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend1_SNoLev
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegbdpfdfoepgmefggh} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{\mname{{SNoLev}}}}~{{({{{{\mname{{SNo\_extend1}}}}~{{x}}}})}}}}={{{{\mname{{ordsucc}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 33cf6044be79e5a605f13a1596393129b0521a4bbea91a8e8df65c3029e399ba \\
Pure Prop Address: & TMQdtthxiaHskccMdbmaNfQoxB2zGD8wrG3 \\
Theory Prop Id: & 5293470b79054a91a67c775285d231ecc230d1826978e4d088daf8390581d242 \\
Theory Prop Address: & TMKdFGjuMvwpN2qMjfXbBzHx5ZvwNJUoX9Y
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend0_nIn
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegadpfogjeog} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNoLev}}~{x}}\unicdcdadjd{{\mname{SNo\_extend0}}~{x}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 217cffe317f555501a713a30362da2e786958f5eb0cb0d728c124c2cd996b872 \\
Pure Prop Address: & TMda5fYn6yDo4M7LQx8adXmDoCPdX1T8aKm \\
Theory Prop Id: & 0dfd17ffb30a9a53795a0c34f15c21b135e90759a30b52930e461dc926cfeab8 \\
Theory Prop Address: & TMa2tRaEiFrQbXQteXN4nGzCFeoa2D3wqNJ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend1_In
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegbdpfjeog} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNoLev}}~{x}} \in {{\mname{SNo\_extend1}}~{x}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ac9570cd8593fef418bb2538fbb3c4c48095739f9ed56bba38b9f759f88d20c5 \\
Pure Prop Address: & TMUeMNcoZvQFFpyFh5fDtrPSnekbcwi9gVi \\
Theory Prop Id: & 8ca372c127a4d684d18db02fb51193ea2fd70537694faea2430652d07ea29128 \\
Theory Prop Address: & TMRnfDntKJYsu4fKMmT6ZDjzF5aDKBGdaar
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend0_SNoEq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegadpfdfoepgfebh} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}~{{({{{{\mname{{SNo\_extend0}}}}~{{x}}}})}}}}~{{x}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & bd4a270b5c6aeb7814b4a5a42ab775e0d391605bb207e9a7d4771b5f610877a7 \\
Pure Prop Address: & TMcVbrAftnCJg4KkPuCtBWErWfBTh4ncZk4 \\
Theory Prop Id: & 1299864bcabf2b68aae890ec79e8dac76d30b36a0b72f71d066855a7c6727939 \\
Theory Prop Address: & TMJYn96t53oDn1QogAPqAu66JqLUXXXMPDh
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend1_SNoEq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegbdpfdfoepgfebh} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{SNoEq\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}}~{{({{{{\mname{{SNo\_extend1}}}}~{{x}}}})}}}}~{{x}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ba974f3d548b12b8a092b7b2c74ff719f439552818912e00e7423c8b93a1bd73 \\
Pure Prop Address: & TMF7TV22yna3gp5cUwPCUxbJ6xgGF2dzCxP \\
Theory Prop Id: & d6f8bb86911f71c00536c62a9ff4f7dcb502510ff47cbe6fa6e321ad985965b8 \\
Theory Prop Address: & TMGZEXiMNjzk35x2Tqe624m5WkQJ1kfATtd
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend0_Lt
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegadpfmeeh} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{\mname{SNo\_extend0}}~{x}}<{x}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & bf669a947f56543b6f408b3b0d257dd39de5956a56aa8d486869eea444c8ce02 \\
Pure Prop Address: & TMLC64iAmyHaHX6nQwFVRm3kwcL6vvU4P2u \\
Theory Prop Id: & ea5fd1a237ed14511ef2281ac67e33e3f2c0c0bf15a8e4ea87ffd5dd2258d376 \\
Theory Prop Address: & TMdzN9ZkvjnY1AoibSuChkYwsx6anuPKYr4
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_extend1_Gt
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgihehfgogegbdpfheeh} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{x}<{{\mname{SNo\_extend1}}~{x}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ca0a2d175740921eb82773f6f29e8a9d9ce71765975b1a2abc1810e62321e34d \\
Pure Prop Address: & TMPGx4oU5xXH99g4hKGMXgJyQojSJAw44zk \\
Theory Prop Id: & 1c8d55e5a1c7bc27e06b6ece03d9153ad522999d088945e7671e790cafc9153f \\
Theory Prop Address: & TMSXKAxcPxJwDFbDGV7GdQ6dzrXb8t8Gtfp
\end{tabular}
\end{center}

\end{theorem}
\begin{definition}\label{def:fgahdhpf}
We define $\mname{eps\_}$ to be ${\uniddcgcg n .  {{\{{0}\}}\unicdcdcdbg{\{{\posthc{{({{\mname{ordsucc}}~{m}})}}}|m \in {n}\}}}}$ of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & 5e992a3fe0c82d03dd3d5c5fbd76ae4e09c16d4ce8a8f2bbff757c1617d1cb0b \\
Pure Object Address: & TMPREfrMv4dVM6bYC6yGeUixtCX8s5WtcZH \\
Theory Object Id: & ffee2b2b0138d17826ad706a96fc5a6a6ba8aae8fe09f0db5d0285ae918123b2 \\
Theory Object Address: & TMWgmEV2PgET7GQLdd5raFjcRBfsyaTXCuK
\end{tabular}
\end{center}

\end{definition}
\begin{verbatim}
eps_ordinal_In_eq_0
\end{verbatim}
\begin{theorem}\label{thm:fgahdhpfpgchegjgogbgmgpfjeogpffgbhpfad} ${\unicdcdadad n \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\alpha} \in {{\mname{eps\_}}~{n}}}\unicdbdjdcd{{\alpha}={0}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 3046fd78a51853d1bb9786f97dc51dba530f419189bc9aeb3da38539fa9b461a \\
Pure Prop Address: & TMWe1qmzHkSSwwJFgrwPJoCfonbTQ4efjAN \\
Theory Prop Id: & 1190f379f72b87aef60bf145308c2529adc42bd0db7ec4e1673c896c2a1c59a7 \\
Theory Prop Address: & TMbpDBk3MTcVsKLqZPBPoYrpujcgPfiY5Wv
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
eps_0_1
\end{verbatim}
\begin{theorem}\label{thm:fgahdhpfadpfbd} ${{{\mname{eps\_}}~{0}}={1}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5bbce1fa3c1be09ebf6d116df2665c36b7b8f577503b6c121bb25e865d3b7238 \\
Pure Prop Address: & TMR59LWQ8moWJ5QPeadff1HFh2ASfoVG2gN \\
Theory Prop Id: & d073f03f40b0eb3c8b0bbe33e8eba6928631935514fda8a0b3beeaf04316d10c \\
Theory Prop Address: & TMQ9LV548ZAa1dSucp5vhC4hNL2DmRGxCUz
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo__eps_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfpffgahdhpf} 
$$
\begin{array}{c}
{\unicdcdadad n \in }{{{\mname{{omega}}}}}{ .  }{{{{{{\mname{{SNo\_}}}}~{{({{{{\mname{{ordsucc}}}}~{{n}}}})}}}}~{{({{{{\mname{{eps\_}}}}~{{n}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e9c4d9b771bd18106ae6b62b078902538b29d64f81f15926efc4103f8c00c9ba \\
Pure Prop Address: & TMHkLsYZsJdew3vMRGoV4eMkozdKinHgebF \\
Theory Prop Id: & 1c1428769c5baa199f9145d7fd4da4efa99062ddab3a11093d0b91eaddea75ef \\
Theory Prop Address: & TMZLbVSbviYtp4AZzF9JbpEFvFr85s8f34z
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_eps_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgahdhpf} ${\unicdcdadad n \in {\mname{omega}} .  {{\mname{SNo}}~{({{\mname{eps\_}}~{n}})}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 9fb70039757911dbbd9302330aa8493957a0f839a3c637e3161bffc475cc017a \\
Pure Prop Address: & TMJo1r2k8j9rELnHY45xjWz1j6aXZKmg9xf \\
Theory Prop Id: & 006b0fb6e9891e00a73714a2337f71ea81cd68e1d83e1ffbe6ff6ac2e2ebf23c \\
Theory Prop Address: & TMJdzF6hF6rddoE61FfB1UwP9FVejWKk4Nf
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_eps_
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpffgahdhpf} ${\unicdcdadad n \in {\mname{omega}} .  {{{\mname{SNoLev}}~{({{\mname{eps\_}}~{n}})}}={{\mname{ordsucc}}~{n}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 6891e35f6b0e6661135cf838e9a5fb81c80f47ec3798a2a896ac13c50244ab3b \\
Pure Prop Address: & TMdgBWSxkt84FLUVcqjHhWJ7ceSWs9iyw1B \\
Theory Prop Id: & 75efd151c44e95bbe9b4c51e56bc5088f7629f12f537ea75601bc7ea9eaf088d \\
Theory Prop Address: & TMQsUJFDt5htkcABdDejoFJAbjZMEUCEAeH
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_eps_SNoS_omega
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgahdhpfdfoepgdfpfpgngfghgbg} ${\unicdcdadad n \in {\mname{omega}} .  {{{\mname{eps\_}}~{n}} \in {{\mname{SNoS\_}}~{\mname{omega}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 736d2a5d04c2733a6003f64baced09360d5b4e3fca0641afbbf2823c83ba944a \\
Pure Prop Address: & TMbuXJoMG16dBYE5bnEUW2STmFeuKyFfvUX \\
Theory Prop Id: & 52da1852246ddad0d39ed6113b9f3532b2d250e38f24dfcfeb5bb5b3d673d7d3 \\
Theory Prop Address: & TMWwLLNiXwTEkwv5MiJv3H87ChU7x8J9ACC
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_eps_decr
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgahdhpfegfgdgch} ${\unicdcdadad n \in {\mname{omega}} .  {\unicdcdadad m \in {n} .  {{{\mname{eps\_}}~{n}}<{{\mname{eps\_}}~{m}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d6a46a372ecfabb245c4193957c4f0508d4998eb45deefc85c435af5ad5b3b94 \\
Pure Prop Address: & TMPjtSvp9n2PhtKoxct2FCPEyUEF8mWwigi \\
Theory Prop Id: & 67c76a0faead3dccb86543d7cb1d37298cd531be3091b54b04b83935dc85398b \\
Theory Prop Address: & TMFVwJv6i1e3XzsQSuzx3ek4mzSFTbbAQN6
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_eps_pos
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgahdhpfahpgdh} ${\unicdcdadad n \in {\mname{omega}} .  {{0}<{{\mname{eps\_}}~{n}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8b75de3a77baa9b92adae44b583582eb3b82463db2c14f404402a2bde097e88e \\
Pure Prop Address: & TMYSgyTwMAyFvd1rAZg1g3St39umQ4DAf3o \\
Theory Prop Id: & 760b4aa07773231fa328d11d9a374bf04e0c43853a3127c676e77c310efc55af \\
Theory Prop Address: & TMPXpny51o2j5XrPqyJcP3FHvVjLPiqhf29
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_pos_eps_Lt
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfahpgdhpffgahdhpfmeeh} 
$$
\begin{array}{c}
{\unicdcdadad n .  }{{{{{\mname{{nat\_p}}}}~{{n}}}}}{\unicdbdjdcd}{\unicdcdadad x \in }{{{{{\mname{{SNoS\_}}}}~{{n}}}}}{ .  }{{{{{0}}<{{x}}}}}{\unicdbdjdcd}{{{{{{\mname{{eps\_}}}}~{{n}}}}<{{x}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 575b837b657e6e7a9b06a3cf27feee72addd6cc960da74d3e2bbceddc997f016 \\
Pure Prop Address: & TMXiEk8xFJoDwT18FPGUHmWfruXdJYK5VR8 \\
Theory Prop Id: & d115f0869d3b5e92d6a510264e9d3861cd7160f6be7616adc5574ddeffb04570 \\
Theory Prop Address: & TMMiCk9Z6f7NptPRQzK4m6JDrmCyFfYUM7L
\end{tabular}
\end{center}

\end{theorem}
\section{Ordinals are Surreal Numbers} % TaggedSets2
\begin{verbatim}
ordinal_SNo
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepg} ${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{\mname{SNo}}~{\alpha}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ac39bc678da7d4e002759c32fe0757799548274fcde56c0639a1d25743d1fce3 \\
Pure Prop Address: & TMV1VGX8kBqSXJHN8BCPYuKwoW4HYVoSf7C \\
Theory Prop Id: & 1eea33bee9fd0b26c50050d87b178445fa7eadba2d0a37fed8f158dba3c91ff0 \\
Theory Prop Address: & TMe1KCf7jS8uH1DVEWLEFPafjhTXSfbJGKi
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNoLev
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgmefggh} ${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{SNoLev}}~{\alpha}}={\alpha}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 56f8b04f51175b2d3ad3ec438b868e145d9fb2c1d306f23dedc2aac5944efcc6 \\
Pure Prop Address: & TMZ8sa3eLx4fHfcaC1ehbCESTMyVoJwHZet \\
Theory Prop Id: & fea5fc7aa601d4080a114440a967755767fa47253a68cef3729d30cee0317500 \\
Theory Prop Address: & TMd5j8WdJGFuG8mnG6FCTaEsK9H3VxS3rpz
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNoLev_max
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgmefgghpfngbgih} ${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad z .  {{{\mname{SNo}}~{z}}\unicdbdjdcd{{{{\mname{SNoLev}}~{z}} \in {\alpha}}\unicdbdjdcd{{z}<{\alpha}}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 945794b25c51118cbb7056a200fc29b3beac4a0954ff35c7d46b8b10d9b490eb \\
Pure Prop Address: & TMW5wuDBjEPgptrHgu3NBPzWzTbiLpoMHFG \\
Theory Prop Id: & 51ee47d6504231b0399ea5f21fb49d3219caefd43f8b7ed5b1fa6612bbd4856e \\
Theory Prop Address: & TMMaysrC6GCnCvik13kVigRPVk8Z3bWgCDj
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_In_SNoLt
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfjeogpfdfoepgmeeh} ${\unicdcdadad \alpha .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{\unicdcdadad \beta \in {\alpha} .  {{\beta}<{\alpha}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e37d00b84165adfad6c68f140a3928d30ae38bcb8bc0d296cd08e27ee667c03c \\
Pure Prop Address: & TMKf8Fadm6Fho8o69tQZDnNutgimZmGKfvS \\
Theory Prop Id: & ced78e7d1eec9d129092ce29e5de303b7466ca2b9aed7d07aa7b3a24c939f5c3 \\
Theory Prop Address: & TMX6tHgRu4ztsUaXjsy7UkVCDdBhk7fa6MQ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_SNoLev_max_2
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdfoepgmefgghpfngbgihpfcd} 
$$
\begin{array}{c}
{\unicdcdadad \alpha .  }{{{{{\mname{{ordinal}}}}~{{\alpha}}}}}{\unicdbdjdcd}{\unicdcdadad z .  }{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{{{{{{{\mname{{SNoLev}}}}~{{z}}}} \in {{{{\mname{{ordsucc}}}}~{{\alpha}}}}}}}{\unicdbdjdcd}{{{{z}}\unicdcdgded{{\alpha}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 72c5eb9b297f67eda46b4ae46e71bb272a5437c076df002405f197e27453a158 \\
Pure Prop Address: & TMdSZyUHH9SYckxVb8GyE6S3rr2iCAbjF1T \\
Theory Prop Id: & c9a7312092068758770432c6a1ad0f802c09d7c8bd89c0d980c3a4006485a0bc \\
Theory Prop Address: & TMLopX51i5ivHHzxeprUDyP4DpoUWZeRmxK
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
ordinal_Subq_SNoLe
\end{verbatim}
\begin{theorem}\label{thm:pgchegjgogbgmgpfdffhcgbhpfdfoepgmefg} ${\unicdcdadad \alpha \beta .  {{{\mname{ordinal}}~{\alpha}}\unicdbdjdcd{{{\mname{ordinal}}~{\beta}}\unicdbdjdcd{{{\alpha} \subseteq {\beta}}\unicdbdjdcd{{\alpha}\unicdcdgded{\beta}}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7558ba76794eacd9296839fd46833c13e08c174ebbbd233a9e7e99572119dc6f \\
Pure Prop Address: & TMYpoMN7TNv38QhXrPvAmv9XN6Jwf9myViH \\
Theory Prop Id: & 15c58afcaec3772f6849ca921cc76f9d5a42f0a409f34f90856359657725bb35 \\
Theory Prop Address: & TMYJsKJKFxZReCyT6P9ehV99SK24YQvYPAK
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_etaE
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpffgehbgfe} 
$$
\begin{array}{c}
{\unicdcdadad z .  }{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{\unicdcdadad p : }{{{o}}}{ .  }\\
{{{({{\unicdcdadad L R .  {{{{{{{{\mname{{SNoCutP}}}}~{{L}}}}~{{R}}}}\unicdbdjdcd{{{{({{\unicdcdadad x \in {{L}} .  {{{{{{\mname{{SNoLev}}}}~{{x}}}} \in {{{{\mname{{SNoLev}}}}~{{z}}}}}}}})}}\unicdbdjdcd{{{{({{\unicdcdadad y \in {{R}} .  {{{{{{\mname{{SNoLev}}}}~{{y}}}} \in {{{{\mname{{SNoLev}}}}~{{z}}}}}}}})}}\unicdbdjdcd{{{{{{z}}={{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}}}}\unicdbdjdcd{{p}}}}}}}}}}}})}}}\\
{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & c2447381b74da897eaa58c3af9b1d3dde69eab20d337d33fba3e80ec78a2b254 \\
Pure Prop Address: & TMMaNjK5BsDv7w7K5kZPoSC8eHU2ZpBXnb3 \\
Theory Prop Id: & 109f1d5e9cb018c3887e73bfa9bd86c12f6e0da2cf252e4591c3433b7220949e \\
Theory Prop Address: & TMZNw3WejhC9Mtc8ZSCkmGmtc6te5te6DBB
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_ind
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfjgogeg} 
$$
\begin{array}{c}
{\unicdcdadad P : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }\\
{{{({{\unicdcdadad L R .  {{{{{{{{\mname{{SNoCutP}}}}~{{L}}}}~{{R}}}}\unicdbdjdcd{{{{({{\unicdcdadad x \in {{L}} .  {{{{P}}~{{x}}}}}})}}\unicdbdjdcd{{{{({{\unicdcdadad y \in {{R}} .  {{{{P}}~{{y}}}}}})}}\unicdbdjdcd{{{{P}}~{{({{{{{{\mname{{SNoCut}}}}~{{L}}}}~{{R}}}})}}}}}}}}}}}})}}}{\unicdbdjdcd}{\unicdcdadad z .  }\\
{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{{{{P}}~{{z}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5b4c39e8a208eba4acd367f82c3e27cebe26cfdc469fca07db4499164e4a2ce9 \\
Pure Prop Address: & TMbQGwDAo4c4Qz7pDSP1Qrz23DzxEXC3zuv \\
Theory Prop Id: & 94cbe61705cabfd1ca5e0c74b060f6cbf1387230afda6d17336d57953552fc1a \\
Theory Prop Address: & TMdM7tGeFyUKwEjhWyhtce31kgdVFHS9DFT
\end{tabular}
\end{center}

\end{theorem}
\section{SurrealRecI}\label{dffhchchfgbgmgcffgdgje}Let $ F : {{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{\iota}})}\unicdbdjdcd{\iota}}}$ be given.
\begin{definition}\label{def:dfoepgpfchfgdgpfjg}
$\mname{SNo\_rec\_i}$ is the opaque object of type ${{\iota}\unicdbdjdcd{\iota}}$ identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Object Id: & be45dfaed6c479503a967f3834400c4fd18a8a567c8887787251ed89579f7be3 \\
Pure Object Address: & TMbX65FfYCsMQpKH5CQAuQZ7t7Xzon4LVUj \\
Theory Object Id: & 230ba80e6902fe44eb4a2d0e953e841148a6b8917a639f0a1a7683bf1567f6ed \\
Theory Object Address: & TMFnUMDQ6KoAKr1mcWtFHKKkSNoPUN4jaDz
\end{tabular}
\end{center}

\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}
  \unicdcdadad z .  {{\mname{SNo}}~{z}}\unicdbdjdcd\unicdcdadad g h : {{\iota}\unicdbdjdcd{\iota}} .  {({\unicdcdadad w \in {{\mname{SNoS\_}}~{({{\mname{SNoLev}}~{z}})}} .  {{{g}~{w}}={{h}~{w}}}})}\unicdbdjdcd\\
               {{{{F}~{z}}~{g}}={{{F}~{z}}~{h}}}
\end{array}
\label{hyp23:gech}
\end{equation}
\begin{verbatim}
SNo_rec_i_eq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfchfgdgpfjgpffgbh} 
$$
\begin{array}{c}
{\unicdcdadad z .  }{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{{{{{{\mname{{SNo\_rec\_i}}}}~{{z}}}}={{{{{{F}}~{{z}}}}~{{\mname{{SNo\_rec\_i}}}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d5ffa87a5e29f9bd7db6e5f48c1339a209c1528efb466143479de577f4c925cf \\
Pure Prop Address: & TMFehK7LHhf71LYnf7ptxSG8U7UPEUDMtLB \\
Theory Prop Id: & f547c8e0e249495edd7c8c51106c6527c237807df09c30ce4aae1861b7fd8a48 \\
Theory Prop Address: & TMWhuu65bzKbkqnknDmtEdbETx8pBAhrZni
\end{tabular}
\end{center}

\end{theorem}
\section{SurrealRecII}\label{dffhchchfgbgmgcffgdgjeje}Let $ F : {{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{({{\iota}\unicdbdjdcd{\iota}})}})}\unicdbdjdcd{({{\iota}\unicdbdjdcd{\iota}})}}}$ be given.
\begin{definition}\label{def:dfoepgpfchfgdgpfjgjg}
$\mname{SNo\_rec\_ii}$ 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: & e148e62186e718374accb69cda703e3440725cca8832aed55c0b32731f7401ab \\
Pure Object Address: & TMQRggiyUu4JmM1iVoMTTJfKCnuvGKk9Coq \\
Theory Object Id: & c3f21860c6b33b173235815c3bb495555ca57af947caf72a4ecda2bb7534873b \\
Theory Object Address: & TMSBXU8YNhPPW34DsxuTsKMzbSfmRAeMJrj
\end{tabular}
\end{center}

\end{definition}
Assume the following.
\begin{equation}
  \begin{array}{c}
    \unicdcdadad z .  {{\mname{SNo}}~{z}}\unicdbdjdcd\unicdcdadad g h : {{\iota}\unicdbdjdcd{({{\iota}\unicdbdjdcd{\iota}})}} .  {({\unicdcdadad w \in {{\mname{SNoS\_}}~{({{\mname{SNoLev}}~{z}})}} .  {{{g}~{w}}={{h}~{w}}}})}\unicdbdjdcd\\
                 {{{{F}~{z}}~{g}}={{{F}~{z}}~{h}}}
  \end{array}
  \label{hyp24:gech}
\end{equation}
\begin{verbatim}
SNo_rec_ii_eq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfchfgdgpfjgjgpffgbh} 
$$
\begin{array}{c}
{\unicdcdadad z .  }{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{{{{{{\mname{{SNo\_rec\_ii}}}}~{{z}}}}={{{{{{F}}~{{z}}}}~{{\mname{{SNo\_rec\_ii}}}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 39e866e1054ce982b862417ffa3743bc0e8c324d6dd60d1efde2ee400235eb8c \\
Pure Prop Address: & TMNXn2h6mjf3y4hBzx8Mu2JUFAE9hznp2cd \\
Theory Prop Id: & 1c5373d70e7017dcc19aae17ae9fa209e30fb1e64e38a8cf591d94eb9a8e2f3d \\
Theory Prop Address: & TMHAFMpf4o9ixm66ZLoJTKWUsAPeGaCat6j
\end{tabular}
\end{center}

\end{theorem}
\section{SurrealRec2}\label{dffhchchfgbgmgcffgdgcd}Let $ F : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{{({{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}})}\unicdbdjdcd{\iota}}}}$ be given.
\begin{definition}\label{def:dfoepgpfchfgdgcd}
$\mname{SNo\_rec2}$ 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: & 7d10ab58310ebefb7f8bf63883310aa10fc2535f53bb48db513a868bc02ec028 \\
Pure Object Address: & TMJMVhuN9vEa6QPsu5PJo6CBxM3Wg6MgxXd \\
Theory Object Id: & 9511104c67c657c2ef57d259b59e1c5cb547d2196f59208de8f81071fa6bc63e \\
Theory Object Address: & TMaJ9EqUKsqh8zZDNMETchCY8mQJMz7evJt
\end{tabular}
\end{center}

\end{definition}
Assume the following.
\begin{equation}
  \begin{array}{c}
    \unicdcdadad w .  {{\mname{SNo}}~{w}}\unicdbdjdcd\unicdcdadad z .  {{\mname{SNo}}~{z}}\unicdbdjdcd\\
    \unicdcdadad g h : {{\iota}\unicdbdjdcd{{\iota}\unicdbdjdcd{\iota}}} .  {({\unicdcdadad x \in {{\mname{SNoS\_}}~{({{\mname{SNoLev}}~{w}})}} .  {\unicdcdadad y .  {{{\mname{SNo}}~{y}}\unicdbdjdcd{{{{g}~{x}}~{y}}={{{h}~{x}}~{y}}}}}})}\unicdbdjdcd\\
                 {({\unicdcdadad y \in {{\mname{SNoS\_}}~{({{\mname{SNoLev}}~{z}})}} .  {{{{g}~{w}}~{y}}={{{h}~{w}}~{y}}}})}\unicdbdjdcd\\
                 {{{{{F}~{w}}~{z}}~{g}}={{{{F}~{w}}~{z}}~{h}}}
  \end{array}
  \label{hyp25:gech}
\end{equation}
\begin{verbatim}
SNo_rec2_eq
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfchfgdgcdpffgbh} 
$$
\begin{array}{c}
{\unicdcdadad w .  }{{{{{\mname{{SNo}}}}~{{w}}}}}{\unicdbdjdcd}{\unicdcdadad z .  }{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{{{{{{{{\mname{{SNo\_rec2}}}}~{{w}}}}~{{z}}}}={{{{{{{{F}}~{{w}}}}~{{z}}}}~{{\mname{{SNo\_rec2}}}}}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & e9ea41d7eb248cf4d9445fd415cfe25af80fb49c218714c2380e2789bcae97a8 \\
Pure Prop Address: & TMRDgSHaazow3ek3rzLN6xJFhW5ZdzoekKh \\
Theory Prop Id: & ac9f2b2f651e85145eb6e3ce4dfeb2a9f0bd619baca63e2ceaa4a8bb59e326f5 \\
Theory Prop Address: & TMJGL7o2tFChKYPUZdXzaYEMbNA7fM43fNy
\end{tabular}
\end{center}

\end{theorem}
\section{More Results about Surreal Numbers} % SurrealRec2
\begin{verbatim}
SNo_ordinal_ind
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfpgchegjgogbgmgpfjgogeg} 
$$
\begin{array}{c}
{\unicdcdadad P : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{({{\unicdcdadad \alpha .  {{{{{{\mname{{ordinal}}}}~{{\alpha}}}}\unicdbdjdcd{{\unicdcdadad x \in {{{{\mname{{SNoS\_}}}}~{{\alpha}}}} .  {{{{P}}~{{x}}}}}}}}}})}}}{\unicdbdjdcd}{{({{\unicdcdadad x .  {{{{{{\mname{{SNo}}}}~{{x}}}}\unicdbdjdcd{{{{P}}~{{x}}}}}}}})}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 5b75fbe4f3fb968cae8d18aaf8e357e80492153f9fe23005c02bed666cc4dfac \\
Pure Prop Address: & TMZqP4gVUEbQmwPD8hDsBmcStaLjrwssf66 \\
Theory Prop Id: & dc2c6d6f6bb1c43d5bbae8d635e29154247bd45fc51b459b16c9487e8936aec8 \\
Theory Prop Address: & TMS5emF6ssGYWbwuHaDYXV6uwhf7edEugak
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_ordinal_ind2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfpgchegjgogbgmgpfjgogegcd} 
$$
\begin{array}{c}
{\unicdcdadad P : }{{{{{\iota}}\unicdbdjdcd{{{{\iota}}\unicdbdjdcd{{o}}}}}}}{ .  }\\
{{{({{\unicdcdadad \alpha .  {{{{{{\mname{{ordinal}}}}~{{\alpha}}}}\unicdbdjdcd{{\unicdcdadad \beta .  {{{{{{\mname{{ordinal}}}}~{{\beta}}}}\unicdbdjdcd{{\unicdcdadad x \in {{{{\mname{{SNoS\_}}}}~{{\alpha}}}} .  {{\unicdcdadad y \in {{{{\mname{{SNoS\_}}}}~{{\beta}}}} .  {{{{{{P}}~{{x}}}}~{{y}}}}}}}}}}}}}}}})}}}\\
{\unicdbdjdcd}{{({{\unicdcdadad x y .  {{{{{{\mname{{SNo}}}}~{{x}}}}\unicdbdjdcd{{{{{{\mname{{SNo}}}}~{{y}}}}\unicdbdjdcd{{{{{{P}}~{{x}}}}~{{y}}}}}}}}}})}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ba3315e237ce1d4df16c16a8bf756e2bde89de0b18851b815b96450f882c55c7 \\
Pure Prop Address: & TMJp9FY8VctR1fkJgGYxDdh1RLMeBqeS99x \\
Theory Prop Id: & 4cd5acb49460e8a170f2661eb1292d8e3541c9317544fd5f2d82c4686e8763f0 \\
Theory Prop Address: & TMN4ezAqsQi2C1fr3tcPbKS8LNDQoV38DBY
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_ordinal_ind3
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfpgchegjgogbgmgpfjgogegdd} 
$$
\begin{array}{c}
{\unicdcdadad P : }{{{{{\iota}}\unicdbdjdcd{{{{\iota}}\unicdbdjdcd{{{{\iota}}\unicdbdjdcd{{o}}}}}}}}}{ .  }\\
(\unicdcdadad \alpha .  {{{{\mname{{ordinal}}}}~{{\alpha}}}}\unicdbdjdcd\unicdcdadad \beta .  {{{{\mname{{ordinal}}}}~{{\beta}}}}\unicdbdjdcd\unicdcdadad \gamma .  {{{{\mname{{ordinal}}}}~{{\gamma}}}}\unicdbdjdcd\\
{{\unicdcdadad x \in {{{{\mname{{SNoS\_}}}}~{{\alpha}}}} .  {{\unicdcdadad y \in {{{{\mname{{SNoS\_}}}}~{{\beta}}}} .  {{\unicdcdadad z \in {{{{\mname{{SNoS\_}}}}~{{\gamma}}}} .  {{{{{{{{P}}~{{x}}}}~{{y}}}}~{{z}}}}}}}}}})\\
{\unicdbdjdcd}{{({{\unicdcdadad x y z .  {{{{{{\mname{{SNo}}}}~{{x}}}}\unicdbdjdcd{{{{{{\mname{{SNo}}}}~{{y}}}}\unicdbdjdcd{{{{{{\mname{{SNo}}}}~{{z}}}}\unicdbdjdcd{{{{{{{{P}}~{{x}}}}~{{y}}}}~{{z}}}}}}}}}}}})}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fccdb6672f955c6006797333286f8360f778fdb5570223fd20d9a063212269c4 \\
Pure Prop Address: & TMHEmrGgS81dQJ2JGimKGXSByW4T5buGsKL \\
Theory Prop Id: & dece8cbbffb58ddf8cbf8dcf83ca0a96a6ef9a5448651a114c5355e9cdb4f7d8 \\
Theory Prop Address: & TMSHTjXpVeYmg8pivxYP7m7ifgZn67S1TeJ
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_ind
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpfjgogeg} 
$$
\begin{array}{c}
{\unicdcdadad P : }{{{{{\iota}}\unicdbdjdcd{{o}}}}}{ .  }{{{({{\unicdcdadad x .  {{{{{{\mname{{SNo}}}}~{{x}}}}\unicdbdjdcd{{{{({{\unicdcdadad w \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}} .  {{{{P}}~{{w}}}}}})}}\unicdbdjdcd{{{{P}}~{{x}}}}}}}}}})}}}{\unicdbdjdcd}\\
{{({{\unicdcdadad x .  {{{{{{\mname{{SNo}}}}~{{x}}}}\unicdbdjdcd{{{{P}}~{{x}}}}}}}})}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 8b2b421134af1c1c1d16aec0622c8efee4fd7ec43176635e591cc48df421e73b \\
Pure Prop Address: & TMdjvtesifW1vMYwLzwUFB8ubFQRxeDErX8 \\
Theory Prop Id: & 851a886ebf6c314dbc8a05caeee024ca4081f1783f55486fc9481e9aa423b8ed \\
Theory Prop Address: & TMXG4oEhSvYEBk3fcCYLLydMW8jZPQjHCwg
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_ind2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpfjgogegcd} 
$$
\begin{array}{c}
{\unicdcdadad P : }{{{{{\iota}}\unicdbdjdcd{{{{\iota}}\unicdbdjdcd{{o}}}}}}}{ .  }\\
(\unicdcdadad x y .  {{{{\mname{{SNo}}}}~{{x}}}}\unicdbdjdcd{{{{\mname{{SNo}}}}~{{y}}}}\unicdbdjdcd{{({{\unicdcdadad w \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}} .  {{{{{{P}}~{{w}}}}~{{y}}}}}})}}\unicdbdjdcd\\
{{({{\unicdcdadad z \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}} .  {{{{{{P}}~{{x}}}}~{{z}}}}}})}}\unicdbdjdcd\\
{{{{({{\unicdcdadad w \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}} .  {{\unicdcdadad z \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}} .  {{{{{{P}}~{{w}}}}~{{z}}}}}}}})}}\unicdbdjdcd{{{{{{P}}~{{x}}}}~{{y}}}}}})\\
{\unicdbdjdcd}{\unicdcdadad x y .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{{P}}~{{x}}}}~{{y}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 87eb9a4029d2214aeadffe89e3f00b768dc1f0ee672b65aa78beff4546554a8a \\
Pure Prop Address: & TMPRSJVgigZNu3Ln5T8fN87EJE6rPBh6eyv \\
Theory Prop Id: & 800d4cef7892d2b9082248440b178c0db19d92fa16e49449e7934ce2a5cffeed \\
Theory Prop Address: & TMdh3vKvQM9UhZmdEnwz2CChQjV8xXPGZtD
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_ind3
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpfjgogegdd} 
$$
\begin{array}{c}
{\unicdcdadad P : }{{{{{\iota}}\unicdbdjdcd{{{{\iota}}\unicdbdjdcd{{{{\iota}}\unicdbdjdcd{{o}}}}}}}}}{ .  }\\
(\unicdcdadad x y z .  {{{{\mname{{SNo}}}}~{{x}}}}\unicdbdjdcd{{{{\mname{{SNo}}}}~{{y}}}}\unicdbdjdcd{{{{\mname{{SNo}}}}~{{z}}}}\unicdbdjdcd\\
{{({{\unicdcdadad u \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}} .  {{{{{{{{P}}~{{u}}}}~{{y}}}}~{{z}}}}}})}}\unicdbdjdcd\\
{{({{\unicdcdadad v \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}} .  {{{{{{{{P}}~{{x}}}}~{{v}}}}~{{z}}}}}})}}\unicdbdjdcd\\
{{({{\unicdcdadad w \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{z}}}})}}}} .  {{{{{{{{P}}~{{x}}}}~{{y}}}}~{{w}}}}}})}}\unicdbdjdcd\\
{{({{\unicdcdadad u \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}} .  {{\unicdcdadad v \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}} .  {{{{{{{{P}}~{{u}}}}~{{v}}}}~{{z}}}}}}}})}}\unicdbdjdcd\\
{{({{\unicdcdadad u \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}} .  {{\unicdcdadad w \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{z}}}})}}}} .  {{{{{{{{P}}~{{u}}}}~{{y}}}}~{{w}}}}}}}})}}\unicdbdjdcd\\
{{({{\unicdcdadad v \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}} .  {{\unicdcdadad w \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{z}}}})}}}} .  {{{{{{{{P}}~{{x}}}}~{{v}}}}~{{w}}}}}}}})}}\unicdbdjdcd\\
{{({{\unicdcdadad u \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{x}}}})}}}} .  {{\unicdcdadad v \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{y}}}})}}}} .  {{\unicdcdadad w \in {{{{\mname{{SNoS\_}}}}~{{({{{{\mname{{SNoLev}}}}~{{z}}}})}}}} .  {{{{{{{{P}}~{{u}}}}~{{v}}}}~{{w}}}}}}}}}})}}\unicdbdjdcd\\
   {{{{{{{{P}}~{{x}}}}~{{y}}}}~{{z}}}})\\
{\unicdbdjdcd}{\unicdcdadad x y z .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{y}}}}}{\unicdbdjdcd}{{{{{\mname{{SNo}}}}~{{z}}}}}{\unicdbdjdcd}{{{{{{{{P}}~{{x}}}}~{{y}}}}~{{z}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f1d779fd469c0845a98e0f93f05143afba1386516af9d4603637aac985cd04d3 \\
Pure Prop Address: & TMSvcfsKu6jonEsCbEr8VWi4pk11xZ363HY \\
Theory Prop Id: & ee16b90e02757ceeb6643978af1b3c0c4b10662feeea83cd645e3f428caa2f4e \\
Theory Prop Address: & TMNAmAH2wAfd64peNfnqHxL8gBKfNxd1haR
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_1
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfbd} ${{\mname{SNo}}~{1}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & ac50d79d994bfab93af900919749bb0b00bab77dd02496312cb99636c84de723 \\
Pure Prop Address: & TMLD7rYjwi3XCzS9wBFoZquU2fwKZu8hJfp \\
Theory Prop Id: & 9a733a1814c1f8aa807a0905095d4095b6b6c16bd5c1cbe8d42ff51d0476728e \\
Theory Prop Address: & TMV4EbhEF9Fv43TrwcGbkmuL2KWwTbyEHzw
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfcd} ${{\mname{SNo}}~{2}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 487591f799d85ccddbcd2b6f2e71c42ada69163cc874f37e171eabf04a2cf2c8 \\
Pure Prop Address: & TMLrvp9CHQ6VVsuUey7NKkjK9sekHgGeozC \\
Theory Prop Id: & c524a421cfe10bd6c73ba8913472e276988f04c927f9d815d647884e770ef7e5 \\
Theory Prop Address: & TMR8t9mAk1WDjyns3E36LrbyZ4oDbwT7ZHW
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNo_omega
\end{verbatim}
\begin{theorem}\label{thm:dfoepgpfpgngfghgbg} ${{\mname{SNo}}~{\mname{omega}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & f6584889c3b559c0c57d54079ab6b3db3900f8646787147aba128aae330d55b6 \\
Pure Prop Address: & TMNCVbGvT6eb68ZqscmNFNvebebAFmqdReB \\
Theory Prop Id: & db7b5f8d12a3f6e404156d15406fde7ccaacb30fcba4c04ca7993e10e5bc5231 \\
Theory Prop Address: & TMGovFDv31hbwMQHyVzGRJD2AUNGQ55Jbau
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLt_0_1
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehpfadpfbd} ${{0}<{1}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & dd604718e43f44863c2707c951e0869b557b6fa3a70647ba47a17ec52b0029a2 \\
Pure Prop Address: & TMdgV1jBe6EnZgcQwB7jjy19eTiHyJdqRgk \\
Theory Prop Id: & cb6392897bc04cb1365085f9f877dc153c53b66befbecc65b224bd4cc2f5ecd5 \\
Theory Prop Address: & TMJwENHJCRTJn9GM5HVohBSxeqaDYfTPidF
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLt_0_2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehpfadpfcd} ${{0}<{2}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 7be36fc38585a8b130ffed59f4b86612f5a054fb66623f2e4759235722d406ef \\
Pure Prop Address: & TMaz8fiLqm9K7Mq8fWt23tudQikDLE3nVPq \\
Theory Prop Id: & f469347bc0b459c89c083e9dfd7070a8dfb29072eefa324603e21760ae54f647 \\
Theory Prop Address: & TMHYveHQDjM9W8DaiRmfzZeA4rimU2E92Nx
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLt_1_2
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmeehpfbdpfcd} ${{1}<{2}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & cc44e8f0924d8a19e44142d916783f27b6cc875dba07d11a95a303ffd59e0426 \\
Pure Prop Address: & TMUn8b7nVzm61CdAFH2Td6XCY5QtWtzLVFN \\
Theory Prop Id: & 19d5e1b76af23ecf0247d684e971a9c96fb452c3323f62c2d67d14b22fa104d1 \\
Theory Prop Address: & TMQQHPaJv8GbFZhqEM1fStxij59TT2UfDi7
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
SNoLev_0_eq_0
\end{verbatim}
\begin{theorem}\label{thm:dfoepgmefgghpfadpffgbhpfad} ${\unicdcdadad x .  {{{\mname{SNo}}~{x}}\unicdbdjdcd{{{{\mname{SNoLev}}~{x}}={0}}\unicdbdjdcd{{x}={0}}}}}$. The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & fe0bc4ac89543a28bb90f488d7033f084538a78425338f1c33f2d0f1095b8658 \\
Pure Prop Address: & TMLtxEcKxjJ93DcQ4fddYfsYH8i24duAMQB \\
Theory Prop Id: & dfcf39481eba0eed8e498b098ea072eb9bf52e6e184da7a902a0d3cf7d28b799 \\
Theory Prop Address: & TMQQUQVuJAU1SouCAxFzxgEkDK6uVzMBbqL
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
restr_SNo_
\end{verbatim}
\begin{theorem}\label{thm:chfgdhehchpfdfoepgpf} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha \in }{{{{{\mname{{SNoLev}}}}~{{x}}}}}{ .  }{{{{{{\mname{{SNo\_}}}}~{{\alpha}}}}~{{({{{{x}}\unicdcdcdjd{{{{\mname{{SNoElts\_}}}}~{{\alpha}}}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & d784442182c6268660ac7e2ebd4ce1f49daf2eaccc2ca06df86cf30b983f10b3 \\
Pure Prop Address: & TMKL6VTKgr5Lj5NeDnwUjkGyX4ocPdGDcir \\
Theory Prop Id: & e0eca8857b82373a353cb7cad078aeafe8a30a60da8a14c48961e659c679e16d \\
Theory Prop Address: & TMXSGP5nqZiFAfXN6HwbUUCGasSdVszy7Vr
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
restr_SNo
\end{verbatim}
\begin{theorem}\label{thm:chfgdhehchpfdfoepg} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha \in }{{{{{\mname{{SNoLev}}}}~{{x}}}}}{ .  }{{{{\mname{{SNo}}}}~{{({{{{x}}\unicdcdcdjd{{{{\mname{{SNoElts\_}}}}~{{\alpha}}}}}})}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 930b5633114b598786190b7f3e7477c4a66a906c5e9f9eba7449686faeef6a32 \\
Pure Prop Address: & TMHnTUT8yBBnGwav9Bnqhxz5kqA5ggvvevz \\
Theory Prop Id: & d141ededdd7718b828cf689076cf982e403f154afe9f7dc65662f9704bddf669 \\
Theory Prop Address: & TMSgjXpo7LeuFty8KqVjACKW1eSX1X9yniR
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
restr_SNoLev
\end{verbatim}
\begin{theorem}\label{thm:chfgdhehchpfdfoepgmefggh} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha \in }{{{{{\mname{{SNoLev}}}}~{{x}}}}}{ .  }{{{{{{\mname{{SNoLev}}}}~{{({{{{x}}\unicdcdcdjd{{{{\mname{{SNoElts\_}}}}~{{\alpha}}}}}})}}}}={{\alpha}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & cae8b53a142dc9bd4d328e356d5ea30fde7a5099b334cda244e99187209b8b99 \\
Pure Prop Address: & TMS5WZkxehafSgkaqhUw2VJHacUTw9rwS1L \\
Theory Prop Id: & 8b18538bf7d6bd1d9a848a9a9517a642deb5a218ec3d8a7caacd524d21dfd64a \\
Theory Prop Address: & TMJN6jAeeeX5njqn5SeaWsGAgEsFLV3XR4u
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
restr_SNoEq
\end{verbatim}
\begin{theorem}\label{thm:chfgdhehchpfdfoepgfebh} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha \in }{{{{{\mname{{SNoLev}}}}~{{x}}}}}{ .  }{{{{{{{{\mname{{SNoEq\_}}}}~{{\alpha}}}}~{{({{{{x}}\unicdcdcdjd{{{{\mname{{SNoElts\_}}}}~{{\alpha}}}}}})}}}}~{{x}}}}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 829f8b1631e1372f4e65a822476f355e1c77a81a384b9cca1a4a8d552190fc4d \\
Pure Prop Address: & TMRgDXhz8Afh4H4FduKLm7tdh5mLBCuj1VX \\
Theory Prop Id: & 704f5aec2f423db02021a4327ff03449e11de8691265c76b45d8784294ccc34c \\
Theory Prop Address: & TMZQtNrbPSFdXyaV4jvd6yZ7nzE191PFXRr
\end{tabular}
\end{center}

\end{theorem}
\begin{verbatim}
restr_SNo_SNoCut
\end{verbatim}
\begin{theorem}\label{thm:chfgdhehchpfdfoepgpfdfoepgdefheh} 
$$
\begin{array}{c}
{\unicdcdadad x .  }{{{{{\mname{{SNo}}}}~{{x}}}}}{\unicdbdjdcd}{\unicdcdadad \alpha \in }{{{{{\mname{{SNoLev}}}}~{{x}}}}}{ .  }{\unicdcdadad p : }{{{o}}}{ .  }\\
({{{{{{\mname{{SNoCutP}}}}~{{\{w \in {{{{\mname{{SNoL}}}}~{{x}}}}|{{{{{{\mname{{SNoLev}}}}~{{w}}}} \in {{\alpha}}}}\}}}}}~{{\{z \in {{{{\mname{{SNoR}}}}~{{x}}}}|{{{{{{\mname{{SNoLev}}}}~{{z}}}} \in {{\alpha}}}}\}}}}}\unicdbdjdcd\\
{{{{{{x}}\unicdcdcdjd{{{{\mname{{SNoElts\_}}}}~{{\alpha}}}}}}={{{{{{\mname{{SNoCut}}}}~{{\{w \in {{{{\mname{{SNoL}}}}~{{x}}}}|{{{{{{\mname{{SNoLev}}}}~{{w}}}} \in {{\alpha}}}}\}}}}}~{{\{z \in {{{{\mname{{SNoR}}}}~{{x}}}}|{{{{{{\mname{{SNoLev}}}}~{{z}}}} \in {{\alpha}}}}\}}}}}}}\\
\unicdbdjdcd{{p}})\\
{\unicdbdjdcd}{p}.
\end{array}
$$
 The proposition is identified by the following information:
\begin{center}
\scriptsize
\begin{tabular}{rl}
Pure Prop Id: & 89a30eaccf26f7f6a96eaa958a6ba761b57a0f92bbecd3515a4233b677745527 \\
Pure Prop Address: & TMEjpjmejvYuUCvTudHxbUChhzKXViX36A4 \\
Theory Prop Id: & 965ec873f3ef2f209dcc57482d884d08d8c7e0ccc428c255bb178794d09f6adc \\
Theory Prop Address: & TMJDSz4hS22ToFR3pnjsU5VSzcSsLte1yVF
\end{tabular}
\end{center}

\end{theorem}
