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<b>let </b><font color="Maroon" title="c1">S</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><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>;<br/><b>let </b><font color="Maroon" title="c3">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>;<br/><b>let </b><font color="Maroon" title="c4">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>;<br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c3">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="c4">ts</font>
 ;<br/>

<b>set </b><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/>
<b>set </b><font color="Maroon" title="c6">PTA</font> =  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<a NAME="E2:29"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c3">nt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <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> :  ex <font color="Olive" title="b2">n</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <font color="Olive" title="b1">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">n</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="osafree.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:29"/>
<font color="Maroon" title="c3">nt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K7" title="DTCONSTR:func.7">NonTerminals</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>
 <b>by </b><i><a class="ref" href="lang1.html#D3" title="LANG1:def.3">LANG1:def 3</a></i>;<br/>

<a NAME="E4:29"/><b>then </b>
<font color="Maroon" title="c3">nt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">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#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="osafree.html#T3" target="_self" title="OSAFREE:th.3">Th3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c8">b1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:29"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c7">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c3">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="c7">o1</font>,<font color="Maroon" title="c8">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">o1</font> = <font color="Maroon" title="c7">o1</font> as   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="osafree.html#E6:29"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">nt1</font> = <font color="Maroon" title="c3">nt</font> as   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</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> <b>by </b><i><a class="txt" href="osafree.html#E2:29"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="lang1.html#D3" title="LANG1:def.3">LANG1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">ts1</font> = <font color="Maroon" title="c4">ts</font> as    <a href="dtconstr.html#M2" title="DTCONSTR:mode.2">SubtreeSeq</a> of <font color="Maroon" title="c10">nt1</font> <b>by </b><i><a class="txt" href="osafree.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="dtconstr.html#D9" title="DTCONSTR:def.9">DTCONSTR:def 9</a></i>;<br/>
<a NAME="E10:29"/>
<font color="Maroon" title="c10">nt1</font> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c11">ts1</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>
 
;<br/>
<b>hence </b><a NAME="E11:29"/>
<font color="Maroon" title="c3">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c4">ts</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>
 ;<br/>

<a NAME="E12:29"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c8">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E6:29"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">o1</font>
;<br/>

<b>thus </b><a NAME="E13:29"/>
<font color="Maroon" title="c3">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="c9">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>
 <b>by </b><i><a class="txt" href="osafree.html#E6:29"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>

<a NAME="E14:29"/><i><font color="Green" title="E21">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">nt1</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c4">ts</font></span>)</span> <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="c9">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>
 
<b>by </b><i><a class="txt" href="osafree.html#E6:29"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="osafree.html#E12:29"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">p</font> being    <a href="dtconstr.html#M2" title="DTCONSTR:mode.2">SubtreeSeq</a> of  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c9">o1</font>,<font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E16:29"/><i><font color="Green" title="E22">A6</font></i>: 
( <font color="Maroon" title="c10">nt1</font> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-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="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c9">o1</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c12">p</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c9">o1</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="c12">p</font> &amp; <font color="Maroon" title="c12">p</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="c9">o1</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> &amp; <font color="Maroon" title="c10">nt1</font> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-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="c9">o1</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="c12">p</font> )
 <b>by </b><i><a class="ref" href="osafree.html#T11" target="_self" title="OSAFREE:th.11">Th11</a></i>;<br/>
<b>thus </b><a NAME="E17:29"/>
( <font color="Maroon" title="c4">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="c9">o1</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> &amp; <font color="Maroon" title="c3">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c4">ts</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="c9">o1</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="c4">ts</font> &amp; ( 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="c3">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c4">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="c9">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s1</font> ) ) )
 <b>by </b><i><a class="txt" href="osafree.html#E14:29"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="osafree.html#E16:29"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="osafree.html#T11" target="_self" title="OSAFREE:th.11">Th11</a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/>


</div>
