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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="osalg_1.html#V8" title="OSALG_1:attr.8">monotone</a>   <a href="osalg_1.html#V10" title="OSALG_1:attr.10">regular</a>   <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a>  <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">t</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Olive" title="b1">X</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Olive" title="b1">X</font></span>)</span>,<font color="Olive" title="b2">t</font> &amp;  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Olive" title="b2">t</font> &amp; ( for <font color="Olive" title="b3">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">s</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">x</font>,<font color="Olive" title="b3">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> ) &amp; ( for <font color="Olive" title="b3">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b4">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Olive" title="b1">X</font></span>)</span>  st  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b3">o</font>,<font color="Olive" title="b1">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b4">ts</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b3">o</font>,<font color="Olive" title="b1">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b4">ts</font> holds <br/>(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b4">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b3">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b3">o</font>,<font color="Olive" title="b1">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b4">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b3">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b4">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b4">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b3">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b4">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b4">ts</font></span>)</span> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Olive" title="b1">t</font> &amp;  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Olive" title="b1">t</font> &amp; ( for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">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="Olive" title="b2">s</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> ) &amp; ( for <font color="Olive" title="b2">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b2">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">ts</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b2">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">ts</font> holds <br/>(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b3">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b2">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b2">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b3">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b2">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b3">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b3">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b2">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b3">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b3">ts</font></span>)</span> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font> be    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">t</font> &amp;  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c3">t</font> &amp; ( for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> ) &amp; ( for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font> holds <br/>(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> ) ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">PTA</font> =  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">D</font> =  <a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">R</font> =  <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">SPTA</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">M</font> =  <a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">F</font> =  <a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,$1;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <span class="kw">means </span> <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> $1;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <span class="kw">means </span>for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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="Olive" title="b1">s</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <span class="kw">means </span>for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font> holds <br/>(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <span class="kw">means </span> ex <font color="Olive" title="b1">t1</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">t1</font>] );<br/>
<a NAME="E1:86"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b2">ts1</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts1</font> &amp; ( for <font color="Olive" title="b3">dt1</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b3">dt1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">ts1</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b3">dt1</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">ts1</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Maroon" title="c10">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b1">ts1</font> &amp; ( for <font color="Olive" title="b2">dt1</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b2">dt1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">ts1</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">dt1</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b1">ts1</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c11">ts1</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">ts1</font> &amp; ( for <font color="Olive" title="b1">dt1</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">dt1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts1</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">dt1</font>] ) implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:86_1"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon" title="c10">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">ts1</font>
 <span class="kw">and </span><br/><a NAME="E2:86_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
for <font color="Olive" title="b1">dt1</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">dt1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts1</font> holds <br/> ex <font color="Olive" title="b2">t2</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">dt1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t2</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">t2</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">t2</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b2">t2</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">t1</font> = <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></span>;<br/>
<a NAME="E4:86_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">o</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E6:86_1"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
<font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:86_1"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
<font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<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="kw">and </span><br/><a NAME="E8:86_1"/><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">ts1</font>
 <span class="kw">and </span><br/><a NAME="E9:86_1"/>
for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> iff  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c13">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">Ms</font> = <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">LSMs</font> =  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">w</font> =  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">so</font> =  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c13">o</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">Lo</font> =  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span>;<br/>
<a NAME="E10:86_1"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:86_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></span>;<br/>
<a NAME="E11:86_1"/><span class="lab"><font color="Green" title="E67">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></span>;<br/>
<a NAME="E12:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D14" target="_self" title="OSAFREE:def.14">Def14</a></span>;<br/>
<a NAME="E13:86_1"/>
 <a href="osafree.html#K13" title="OSAFREE:func.13">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="osafree.html#D16" target="_self" title="OSAFREE:def.16">Def16</a></span>;<br/>
<a NAME="E14:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A11</font></span>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K12" title="OSAFREE:func.12">pi</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="osafree.html#D32" target="_self" title="OSAFREE:def.32">Def32</a></span>;<br/>
<a NAME="E15:86_1"/><span class="lab"><font color="Green" title="E70">A12</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c12">t1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">o2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b1">ts</font> &amp; <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b1">ts</font> holds <br/>(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o2</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c20">ts</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c20">ts</font> &amp; <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c20">ts</font> implies (  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o2</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:86_1_1"/>
 <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c20">ts</font>
 <span class="kw">and </span><br/><a NAME="E2:86_1_1"/><span class="lab"><font color="Green" title="E71">A13</font></span>: 
<font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c20">ts</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o2</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c21">Ms1</font> = <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">LSMs1</font> =  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">Lo2</font> =  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:86_1_1"/><span class="lab"><font color="Green" title="E72">A14</font></span>: 
<font color="Maroon" title="c20">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ts1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></span>;<br/>
<a NAME="E4:86_1_1"/><span class="lab"><font color="Green" title="E73">A15</font></span>: 
 <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nt</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></span>;<br/>
<a NAME="E5:86_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A16</font></span>: 
<font color="Maroon" title="c19">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E6:86_1_1"/><span class="lab"><font color="Green" title="E75">A17</font></span>: 
 <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:86_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:86_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="txt" href="#E12:86_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="osalg_1.html#D7" title="OSALG_1:def.7">OSALG_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> or for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive">b<sub>3</sub></font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c24">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> or for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:86_1_1_1"/><span class="lab"><font color="Green" title="E76">A18</font></span>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c24">i</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a></span>;<br/>
<a NAME="E4:86_1_1_1"/>
<font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:86_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="txt" href="#E2:86_1_1_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">tr</font> = <font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a></span>;<br/>
<a NAME="E6:86_1_1_1"/><span class="lab"><font color="Green" title="E77">A19</font></span>: 
 ex <font color="Olive" title="b1">tr1</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">tr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">tr</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">tr1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">tr1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">tr1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">tr1</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E12:86_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="txt" href="#E2:86_1_1_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E7:86_1_1_1"/><span class="lab"><font color="Green" title="E78">A20</font></span>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:86_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="txt" href="#E12:86_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="txt" href="#E2:86_1_1_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E8:86_1_1_1"/><span class="lab"><font color="Green" title="E79">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">tr</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:86_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="txt" href="#E12:86_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="txt" href="#E2:86_1_1_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></span>;<br/>
<a NAME="E9:86_1_1_1"/>
<font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:86_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E10:86_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="txt" href="#E12:86_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="txt" href="#E2:86_1_1_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="ref" href="msualg_6.html#T2" title="MSUALG_6:th.2">MSUALG_6:2</a></span>;<br/>
<a NAME="E10:86_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A22</font></span>: 
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c26">tr</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c27">s1</font>, <font color="Maroon" title="c28">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c27">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> or  not <font color="Maroon" title="c28">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> or <font color="Maroon" title="c27">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c28">s2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E11:86_1_1_1"/><span class="lab"><font color="Green" title="E81">A23</font></span>: 
<font color="Maroon" title="c27">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
 <span class="kw">and </span><br/><a NAME="E12:86_1_1_1"/><span class="lab"><font color="Green" title="E82">A24</font></span>: 
<font color="Maroon" title="c28">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c28">s2</font></span><br/>

<a NAME="E13:86_1_1_1"/>
 ex <font color="Olive" title="b1">t3</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">t3</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="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Olive" title="b1">t3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:86_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="txt" href="#E12:86_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="txt" href="#E2:86_1_1_1"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="ref" href="osafree.html#D14" target="_self" title="OSAFREE:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:86_1_1_1"/>
<font color="Maroon" title="c27">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c28">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1_1_1"><span class="lab"><font color="Green" title="E77">A19</font></span></a>, <a class="txt" href="#E11:86_1_1_1"><span class="lab"><font color="Green" title="E81">A23</font></span></a>, <a class="txt" href="#E12:86_1_1_1"><span class="lab"><font color="Green" title="E82">A24</font></span></a>, <a class="txt" href="#E8:86_1_1_1"><span class="lab"><font color="Green" title="E79">A21</font></span></a>, <a class="txt" href="#E10:86_1_1_1"><span class="lab"><font color="Green" title="E80">A22</font></span></a>, <a class="txt" href="#E7:86_1_1_1"><span class="lab"><font color="Green" title="E78">A20</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:86_1_1"/>
 <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E4:86_1_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="txt" href="#E3:86_1_1"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> )</span><br/>

<a NAME="E8:86_1_1"/>
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span> <a href="osalg_1.html#R5" title="OSALG_1:pred.5">has_least_rank_for</a> <font color="Maroon" title="c19">o2</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1_1"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="txt" href="#E5:86_1_1"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="txt" href="#E6:86_1_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="ref" href="osalg_1.html#T14" title="OSALG_1:th.14">OSALG_1:14</a></span>;<br/>
<a NAME="E9:86_1_1"/><span class="kw">then </span>
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c19">o2</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D12" title="OSALG_1:def.12">OSALG_1:def 12</a></span>;<br/>
<a NAME="E10:86_1_1"/><span class="kw">then </span>
 <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D10" title="OSALG_1:def.10">OSALG_1:def 10</a></span>;<br/>
<a NAME="E11:86_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A25</font></span>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<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="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T19" target="_self" title="OSAFREE:th.19">Th19</a></span>;<br/>
<a NAME="E12:86_1_1"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c19">o2</font>,<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="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1_1"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="txt" href="#E5:86_1_1"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="txt" href="#E6:86_1_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="ref" href="osafree.html#T19" target="_self" title="OSAFREE:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:86_1_1"/>
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c19">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:86_1_1"><span class="lab"><font color="Green" title="E76">A25</font></span></a>, <a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E14:86_1_1"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c19">o2</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c20">ts</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:86_1"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="txt" href="#E3:86_1_1"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="txt" href="#E5:86_1_1"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="ref" href="osafree.html#D15" target="_self" title="OSAFREE:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:86_1"/><span class="lab"><font color="Green" title="E71">A26</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c19">s</font> &amp; <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c19">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">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="c19">s</font> &amp; <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:86_1_2"/><span class="lab"><font color="Green" title="E72">A27</font></span>: 
<font color="Maroon" title="c20">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="c19">s</font>
 <span class="kw">and </span><br/><a NAME="E2:86_1_2"/><span class="lab"><font color="Green" title="E73">A28</font></span>: 
<font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font></span><br/>

<a NAME="E3:86_1_2"/>
<font color="Maroon" title="c12">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:86_1_2"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_2"><span class="lab"><font color="Green" title="E72">A27</font></span></a>, <a class="txt" href="#E2:86_1_2"><span class="lab"><font color="Green" title="E73">A28</font></span></a>, <a class="ref" href="osafree.html#T10" target="_self" title="OSAFREE:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E72">A29</font></span>: <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> = 
 <a href="osafree.html#K18" title="OSAFREE:func.18">@</a> <font color="Maroon" title="c10">nt</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="osafree.html#D22" target="_self" title="OSAFREE:def.22">Def22</a></span>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o2</font>,<span class="p5">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font></span>)</span>,<font color="Olive" title="b3">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o2</font> is   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>, <font color="Olive" title="b2">x2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o2</font>,<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>, <font color="Olive" title="b3">s3</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : (  ex <font color="Olive" title="b4">o1</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b1">o2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b4">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b4">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s3</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s3</font> ) &amp;  ex <font color="Olive" title="b4">w3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b4">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> &amp; ( for <font color="Olive" title="b5">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">w3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> ) ) ) </span>}</span> 

;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">t1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c12">t1</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E18:86_1"/>
<font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c12">t1</font>] )</span><br/>

<a NAME="E19:86_1"/><span class="lab"><font color="Green" title="E73">A30</font></span>: 
 ex <font color="Olive" title="b1">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">ta1</font> = <font color="Maroon" title="c12">t1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c12">t1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></span>;<br/>
<a NAME="E21:86_1"/><span class="lab"><font color="Green" title="E74">A31</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></span>;<br/>
<a NAME="E22:86_1"/><span class="lab"><font color="Green" title="E75">A32</font></span>: 
<font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c13">o</font>,<font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a></span>;<br/>
<a NAME="E23:86_1"/><span class="kw">then </span>
 <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E15:86_1"><span class="lab"><font color="Green" title="E70">A12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">o3</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E25:86_1"/><span class="lab"><font color="Green" title="E76">A33</font></span>: 
 <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">o3</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E26:86_1"/><span class="lab"><font color="Green" title="E77">A34</font></span>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c20">o3</font>,<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="kw">and </span><br/><a NAME="E27:86_1"/><span class="lab"><font color="Green" title="E78">A35</font></span>: 
<span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c20">o3</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E28:86_1"/>
for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> iff  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c20">o3</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">Msr</font> = <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<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="kw">by</span> <span class="lab"><a class="txt" href="#E25:86_1"><span class="lab"><font color="Green" title="E76">A33</font></span></a>, <a class="txt" href="#E26:86_1"><span class="lab"><font color="Green" title="E77">A34</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E30:86_1"/><span class="lab"><font color="Green" title="E79">A36</font></span>: 
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">o3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:86_1"><span class="lab"><font color="Green" title="E76">A33</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E31:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A37</font></span>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">Msr</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E22:86_1"><span class="lab"><font color="Green" title="E75">A32</font></span></a>, <a class="txt" href="#E15:86_1"><span class="lab"><font color="Green" title="E70">A12</font></span></a>, <a class="txt" href="#E27:86_1"><span class="lab"><font color="Green" title="E78">A35</font></span></a></span>;<br/>
<a NAME="E32:86_1"/><span class="lab"><font color="Green" title="E81">A38</font></span>: 
 <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E22:86_1"><span class="lab"><font color="Green" title="E75">A32</font></span></a>, <a class="txt" href="#E15:86_1"><span class="lab"><font color="Green" title="E70">A12</font></span></a></span>;<br/>
<a NAME="E33:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A39</font></span>: 
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="osalg_1.html#R5" title="OSALG_1:pred.5">has_least_rank_for</a> <font color="Maroon" title="c13">o</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T14" title="OSALG_1:th.14">OSALG_1:14</a></span>;<br/>
<a NAME="E34:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A40</font></span>: 
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="osalg_1.html#R4" title="OSALG_1:pred.4">has_least_sort_for</a> <font color="Maroon" title="c13">o</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D12" title="OSALG_1:def.12">OSALG_1:def 12</a></span>;<br/>
<a NAME="E35:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A41</font></span>: 
<font color="Maroon" title="c13">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a>  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D11" title="OSALG_1:def.11">OSALG_1:def 11</a></span>;<br/>
<a NAME="E36:86_1"/>
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c13">o</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:86_1"><span class="lab"><font color="Green" title="E82">A39</font></span></a>, <a class="ref" href="osalg_1.html#D12" title="OSALG_1:def.12">OSALG_1:def 12</a></span>;<br/>
<a NAME="E37:86_1"/><span class="kw">then </span>
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:86_1"><span class="lab"><font color="Green" title="E81">A38</font></span></a>, <a class="ref" href="osalg_1.html#D10" title="OSALG_1:def.10">OSALG_1:def 10</a></span>;<br/>
<a NAME="E38:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A42</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D7" title="OSALG_1:def.7">OSALG_1:def 7</a></span>;<br/>
<span class="lab"><font color="Green" title="E86">A43</font></span>:  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c12">t1</font> = 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c19">ta1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></span>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T26" target="_self" title="OSAFREE:th.26">Th26</a></span>
;<br/>
<a NAME="E40:86_1"/><span class="lab"><font color="Green" title="E87">A44</font></span>: 
 <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:86_1"><span class="lab"><font color="Green" title="E81">A38</font></span></a>, <a class="txt" href="#E34:86_1"><span class="lab"><font color="Green" title="E83">A40</font></span></a>, <a class="ref" href="osalg_1.html#D11" title="OSALG_1:def.11">OSALG_1:def 11</a></span>;<br/>
<a NAME="E41:86_1"/><span class="lab"><font color="Green" title="E88">A45</font></span>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c12">t1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,$2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E1:86_1_5"/><span class="lab"><font color="Green" title="E89">A46</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">s4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">s4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">s4</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> implies  ex <font color="Olive" title="b1">s4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c22">i</font>,<font color="Olive" title="b1">s4</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:86_1_5_1"/><span class="lab"><font color="Green" title="E90">A47</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c22">i</font>,<font color="Olive" title="b1">s4</font>] )</span><br/>

<a NAME="E2:86_1_5_1"/><span class="lab"><font color="Green" title="E91">A48</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_5_1"><span class="lab"><font color="Green" title="E90">A47</font></span></a>, <a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></span>;<br/>
<a NAME="E3:86_1_5_1"/>
<font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:86_1"><span class="lab"><font color="Green" title="E74">A31</font></span></a>, <a class="txt" href="#E1:86_1_5_1"><span class="lab"><font color="Green" title="E90">A47</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">td1</font> = <font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a></span>;<br/>
<a NAME="E5:86_1_5_1"/><span class="lab"><font color="Green" title="E92">A49</font></span>: 
 ex <font color="Olive" title="b1">td2</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">td2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">td1</font> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">td2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Olive" title="b1">td2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">td2</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">td2</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">td2</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E21:86_1"><span class="lab"><font color="Green" title="E74">A31</font></span></a>, <a class="txt" href="#E1:86_1_5_1"><span class="lab"><font color="Green" title="E90">A47</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E6:86_1_5_1"/><span class="lab"><font color="Green" title="E93">A50</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:86_1"><span class="lab"><font color="Green" title="E74">A31</font></span></a>, <a class="txt" href="#E1:86_1_5_1"><span class="lab"><font color="Green" title="E90">A47</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">tda</font> = <font color="Maroon" title="c23">td1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c23">td1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></span>;<br/>
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c23">td1</font> = 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c24">tda</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></span>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_5_1"><span class="lab"><font color="Green" title="E91">A48</font></span></a>, <a class="ref" href="osafree.html#T26" target="_self" title="OSAFREE:th.26">Th26</a></span>
;<br/>
<a NAME="E9:86_1_5_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>,<font color="Olive" title="b1">s4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1_5_1"><span class="lab"><font color="Green" title="E93">A50</font></span></a>, <a class="txt" href="#E5:86_1_5_1"><span class="lab"><font color="Green" title="E92">A49</font></span></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:86_1_5_1"/>
 ex <font color="Olive" title="b1">s4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c22">i</font>,<font color="Olive" title="b1">s4</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_5_1"><span class="lab"><font color="Green" title="E91">A48</font></span></a>, <a class="ref" href="osafree.html#T24" target="_self" title="OSAFREE:th.24">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c22">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:86_1_5"/><span class="lab"><font color="Green" title="E90">A51</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="#E1:86_1_5"><span class="lab"><font color="Green" title="E89">A46</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c23">wa</font> = <font color="Maroon" title="c22">f</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:86_1"><span class="lab"><font color="Green" title="E74">A31</font></span></a>, <a class="txt" href="#E19:86_1"><span class="lab"><font color="Green" title="E73">A30</font></span></a>, <a class="txt" href="#E3:86_1_5"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">wa</font> = <font color="Maroon" title="c23">wa</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1_5"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">wa</font> = <font color="Maroon" title="c24">wa</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E7:86_1_5"/>
for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Msr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">wa</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">y</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Msr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">wa</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:86_1_5_2"/><span class="lab"><font color="Green" title="E91">A52</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Msr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">wa</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font></span><br/>

<a NAME="E2:86_1_5_2"/>
<font color="Maroon" title="c25">wa</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">wa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1_5"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="txt" href="#E1:86_1_5_2"><span class="lab"><font color="Green" title="E91">A52</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:86_1_5_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Msr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">wa</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1_5"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="txt" href="#E1:86_1_5_2"><span class="lab"><font color="Green" title="E91">A52</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:86_1_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p4">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p5">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">Msr</font></span>)</span>,<span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E17:86_1"><span class="lab"><font color="Green" title="E72">A29</font></span></a>, <a class="txt" href="#E35:86_1"><span class="lab"><font color="Green" title="E84">A41</font></span></a>, <a class="txt" href="#E40:86_1"><span class="lab"><font color="Green" title="E87">A44</font></span></a>, <a class="txt" href="#E38:86_1"><span class="lab"><font color="Green" title="E85">A42</font></span></a>, <a class="txt" href="#E3:86_1_5"><span class="lab"><font color="Green" title="E90">A51</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:86_1_5"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c12">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:86_1"><span class="lab"><font color="Green" title="E86">A43</font></span></a>, <a class="txt" href="#E31:86_1"><span class="lab"><font color="Green" title="E80">A37</font></span></a>, <a class="ref" href="funct_5.html#T4" title="FUNCT_5:th.4">FUNCT_5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E42:86_1"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E22:86_1"><span class="lab"><font color="Green" title="E75">A32</font></span></a>, <a class="txt" href="#E15:86_1"><span class="lab"><font color="Green" title="E70">A12</font></span></a></span>;<br/>
<a NAME="E43:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A53</font></span>: 
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:86_1"><span class="lab"><font color="Green" title="E77">A34</font></span></a>, <a class="txt" href="#E27:86_1"><span class="lab"><font color="Green" title="E78">A35</font></span></a>, <a class="txt" href="#E30:86_1"><span class="lab"><font color="Green" title="E79">A36</font></span></a>, <a class="ref" href="osafree.html#T18" target="_self" title="OSAFREE:th.18">Th18</a></span>;<br/>
<a NAME="E44:86_1"/>
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c13">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:86_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E8:86_1"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="ref" href="osafree.html#T18" target="_self" title="OSAFREE:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E45:86_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c12">t1</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:86_1"><span class="lab"><font color="Green" title="E70">A12</font></span></a>, <a class="txt" href="#E32:86_1"><span class="lab"><font color="Green" title="E81">A38</font></span></a>, <a class="txt" href="#E43:86_1"><span class="lab"><font color="Green" title="E89">A53</font></span></a>, <a class="txt" href="#E34:86_1"><span class="lab"><font color="Green" title="E83">A40</font></span></a>, <a class="txt" href="#E41:86_1"><span class="lab"><font color="Green" title="E88">A45</font></span></a>, <a class="txt" href="#E16:86_1"><span class="lab"><font color="Green" title="E71">A26</font></span></a>, <a class="ref" href="osalg_1.html#D11" title="OSALG_1:def.11">OSALG_1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:86"/><span class="lab"><font color="Green" title="E60">A54</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">sy</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">sy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> implies <font color="Maroon">S<sub>5</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:86_2"/><span class="lab"><font color="Green" title="E61">A55</font></span>: 
<font color="Maroon" title="c10">sy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">t1</font> =  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_2"><span class="lab"><font color="Green" title="E61">A55</font></span></a>, <a class="ref" href="dtconstr.html#D4" title="DTCONSTR:def.4">DTCONSTR:def 4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">t1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c11">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">t1</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E3:86_2"/>
<font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c11">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">t1</font>] )</span><br/>

<span class="lab"><font color="Green" title="E62">A56</font></span>: <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</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#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font></span>)</span> = 
 <a href="osafree.html#K14" title="OSAFREE:func.14">pi</a> <font color="Maroon" title="c10">sy</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_2"><span class="lab"><font color="Green" title="E61">A55</font></span></a>, <a class="ref" href="osafree.html#D32" target="_self" title="OSAFREE:def.32">Def32</a></span>
<br/>.= 
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_2"><span class="lab"><font color="Green" title="E61">A55</font></span></a>, <a class="ref" href="osafree.html#D17" target="_self" title="OSAFREE:def.17">Def17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:86_2"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c11">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T33" target="_self" title="OSAFREE:th.33">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c11">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">t1</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E6:86_2"/>
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c11">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86_2"><span class="lab"><font color="Green" title="E62">A56</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c11">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">t1</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E7:86_2"/>
for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x1</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="Olive" title="b1">s1</font> &amp; <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86_2"><span class="lab"><font color="Green" title="E62">A56</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">t1</font>]</span><br/>

<a NAME="E8:86_2"/><span class="lab"><font color="Green" title="E63">A57</font></span>: 
 ex <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">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="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c10">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_2"><span class="lab"><font color="Green" title="E61">A55</font></span></a>, <a class="ref" href="osafree.html#T4" target="_self" title="OSAFREE:th.4">Th4</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c12">o</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b1">ts</font> &amp; <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b1">ts</font> holds <br/>(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c12">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c12">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b1">ts</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c13">ts</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c13">ts</font> &amp; <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font> implies (  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c12">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c12">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:86_2_2"/>
 <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c13">ts</font>
 <span class="kw">and </span><br/><a NAME="E2:86_2_2"/><span class="lab"><font color="Green" title="E64">A58</font></span>: 
<font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c12">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c12">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> )</span><br/><a NAME="E3:86_2_2"/>
<font color="Maroon" title="c11">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_2_2"><span class="lab"><font color="Green" title="E64">A58</font></span></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:86_2_2"/>
(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c12">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c12">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c13">ts</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:86_2"><span class="lab"><font color="Green" title="E63">A57</font></span></a>, <a class="ref" href="osafree.html#T10" target="_self" title="OSAFREE:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E3:86"/>
for <font color="Olive" title="b1">dt</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">dt</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="dtconstr.html#S7" title="DTCONSTR:sch.7">DTCONSTR:sch 7</a>(<a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E60">A54</font></span></a>, <a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E59">A1</font></span></a>);<br/></span>
<a NAME="E4:86"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">t1</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">t1</font>] )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:86"/>
( <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">t</font> &amp;  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c3">t</font> &amp; ( for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> ) &amp; ( for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font> holds <br/>(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font></span>)</span> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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