<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E30">Th31</font></span>: <a NAME="T31"><span class="comment"><font color="firebrick">:: HEYTING3:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>)</span>)</span><br/>  for <font color="Olive" title="b4">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> (<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>)</span>)</span>  st <font color="Olive" title="b2">a</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Olive" title="b4">X</font> &amp; <font color="Olive" title="b3">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Olive" title="b4">X</font> holds <br/><font color="Olive" title="b2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Olive" title="b4">X</font></div></div>
