<?xml version="1.0"?>
<div class="add">

<span class="kw">consider </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">a</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E2:192_1_1"/><span class="lab"><font color="Green" title="E95">A1</font></span>: 
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K6" title="ABCMIZ_1:func.6">a_Type</a> 
 <span class="kw">and </span><br/><a NAME="E3:192_1_1"/><span class="lab"><font color="Green" title="E96">A2</font></span>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E4:192_1_1"/>
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K7" title="ABCMIZ_1:func.7">an_Adj</a> 
 <span class="kw">and </span><br/><a NAME="E5:192_1_1"/>
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#D12" target="_self" title="ABCMIZ_1:def.12">Def12</a></span>;<br/>
<a NAME="E6:192_1_1"/>
 <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="c2">m</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></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="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:192_1_1"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="txt" href="#E3:192_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="msafree3.html#T5" title="MSAFREE3:th.5">MSAFREE3:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">mm</font> =  <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="c2">m</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T41" target="_self" title="ABCMIZ_1:th.41">Th41</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">mm</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">mm</font> is  <a href="abcmiz_1.html#V4" title="ABCMIZ_1:attr.4">ground</a>  &amp; <font color="Maroon" title="c4">mm</font> is  <a href="abcmiz_1.html#V9" title="ABCMIZ_1:attr.9">pure</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">p</font> =  <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span></span>)</span>;<br/>
<a NAME="E8:192_1_1"/><span class="lab"><font color="Green" title="E97">A3</font></span>: 
<font color="Maroon" title="c4">mm</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="c2">m</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">C</font>,<span class="p4">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T20" title="TREES_4:th.20">TREES_4:20</a></span>;<br/>
<a NAME="E9:192_1_1"/><span class="lab"><font color="Green" title="E98">A4</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:192_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="abcmiz_1.html#D9" target="_self" title="ABCMIZ_1:def.9">Def9</a></span>;<br/>
<a NAME="E10:192_1_1"/>
<font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:192_1_1"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="ref" href="abcmiz_1.html#D9" target="_self" title="ABCMIZ_1:def.9">Def9</a></span>;<br/>
<a NAME="E11:192_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A5</font></span>: 
<font color="Maroon" title="c2">m</font> is  <a href="abcmiz_1.html#V2" title="ABCMIZ_1:attr.2">constructor</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:192_1_1"><span class="lab"><font color="Green" title="E98">A4</font></span></a>, <a class="ref" href="abcmiz_1.html#D11" target="_self" title="ABCMIZ_1:def.11">Def11</a></span>;<br/>
<a NAME="E12:192_1_1"/>
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c4">mm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="192_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c4">mm</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:192_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c4">mm</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:192_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">C</font>,<span class="p4">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span></span>)</span></span>)</span> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:192_1_1"><span class="lab"><font color="Green" title="E97">A3</font></span></a>, <a class="txt" href="#E11:192_1_1"><span class="lab"><font color="Green" title="E99">A5</font></span></a>, <a class="ref" href="abcmiz_1.html#T88" target="_self" title="ABCMIZ_1:th.88">Th88</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:192_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">and </span><br/><a NAME="E5:192_1_1_1"/><span class="lab"><font color="Green" title="E100">A6</font></span>: 
<font color="Maroon" title="c7">Y</font> <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="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">C</font>,<span class="p4">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span></span>)</span></span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E6:192_1_1_1"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">C</font>,<span class="p4">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:192_1_1_1"><span class="lab"><font color="Green" title="E100">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:192_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:192_1_1"/><span class="kw">then </span>
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c4">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E14:192_1_1"/>
<font color="Maroon" title="c4">mm</font> is  <a href="abcmiz_1.html#V4" title="ABCMIZ_1:attr.4">ground</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T107" target="_self" title="ABCMIZ_1:th.107">Th107</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">mm</font> is  <a href="abcmiz_1.html#V9" title="ABCMIZ_1:attr.9">pure</a> </span><br/>

<a NAME="E15:192_1_1"/>
 ex <font color="Olive" title="b1">t</font> being    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font> st <br/>( <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="Maroon" title="c2">m</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">t</font> is  <a href="abcmiz_1.html#V9" title="ABCMIZ_1:attr.9">pure</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:192_1_1"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="txt" href="#E3:192_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="abcmiz_1.html#T70" target="_self" title="ABCMIZ_1:th.70">Th70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:192_1_1"/>
<font color="Maroon" title="c4">mm</font> is  <a href="abcmiz_1.html#V9" title="ABCMIZ_1:attr.9">pure</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
