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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E33">Th31</font></span>: <a NAME="T31"><span class="comment"><font color="firebrick">:: MIDSP_3:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">m</font> being    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Olive" title="b1">n</font><br/>  for <font color="Olive" title="b3">RAS</font> being    <a href="midsp_3.html#M2" title="MIDSP_3:mode.2">ReperAlgebra</a> of <font color="Olive" title="b1">n</font><br/>  for <font color="Olive" title="b4">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b3">RAS</font><br/>  for <font color="Olive" title="b5">p</font> being   <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b3">RAS</font><br/>  for <font color="Olive" title="b6">W</font> being   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Olive" title="b3">RAS</font><br/>  for <font color="Olive" title="b7">x</font> being   <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b6">W</font>  st <font color="Olive" title="b6">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> (<font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">x</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Olive" title="b6">W</font> <a href="midsp_3.html#K8" title="MIDSP_3:func.8">.</a> (<font color="Olive" title="b4">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b5">p</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">+*</a> (<span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b2">m</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b7">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b2">m</font></span>)</span>)</div></div>
