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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be    <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:21"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>reconsider </b><font color="Maroon">q</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font> as   <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/>
<a NAME="E3:21"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> : <font color="Olive">a</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:21"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_4.html#T10">TREES_4:10</a></i>;<br/>
<a NAME="E5:21"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/>
<a NAME="E6:21"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font><b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">a</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:21_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <b>by </b><i/>;<br/><a NAME="E7:21_1"/>
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><a NAME="E8:21_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E10:21_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="trees_3.html#D15">TREES_3:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><a NAME="E12:21_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:21_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> as   <a href="trees_2.html#NM4">DecoratedTree</a> <b>by </b><i>, <a class="ref" href="trees_3.html#T26">TREES_3:26</a></i>;<br/><b>take </b><font color="Maroon">t</font> = <font color="Maroon">t</font>;<br/><b>thus </b><a NAME="E15:21_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E16:21_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="trees_2.html#K5">|</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , , , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><a NAME="E17:21_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font></span>)</span> <a href="trees_1.html#K4">|</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><a NAME="E18:21_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><a NAME="E19:21_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:21_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> : <font color="Olive">b</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E21:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">t</font> being   <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><a NAME="E8:21"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 ;<br/>

<a NAME="E9:21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> : <font color="Olive">b</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font><b> such that </b><br/><a NAME="E11:21"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">b</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:21"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E15:21"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:21"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E19:21"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E20:21"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E21:21"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="trees_2.html#K5">|</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span>
 
<b>by </b><i>, , <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E22:21"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E23:21"/><b>then </b>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , , <a class="ref" href="trees_3.html#D15">TREES_3:def 15</a></i>;<br/>
<a NAME="E24:21"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> )
 
<b>by </b><i>, <a class="ref" href="trees_2.html#T34">TREES_2:34</a></i>;<br/>
<b>hence </b><a NAME="E25:21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>


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