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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E34">Th31</font></span>: <a NAME="T31"><span class="comment"><font color="firebrick">:: LTLAXIO4:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <br/>  for <font color="Olive" title="b3">P</font> being   <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a>   <a href="ltlaxio3.html#V3" title="LTLAXIO3:attr.3">complete</a>  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a><br/>  for <font color="Olive" title="b4">T</font> being    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Olive" title="b3">P</font>  st <font color="Olive" title="b1">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; (  for <font color="Olive" title="b5">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Olive" title="b4">T</font>  holds  not <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) holds <br/> for <font color="Olive" title="b5">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Olive" title="b4">T</font> holds <br/> ( <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">Q</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <font color="Olive" title="b1">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</div></div>
