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<b>set </b><font color="Maroon" title="c4">NH</font> =  <a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> );<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:67_1_1"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:67_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p3">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>]
 
<b>by </b><i><a class="txt" href="#E2:67_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:67_1_1"/>
not  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="osafree.html#D26" target="_self" title="OSAFREE:def.26">Def26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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