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<div class="add">

<b>let </b><font color="Maroon">D1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">d1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">d2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font>;<br/><b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="trees_3.html#NM8">DTree-set</a> of <font color="Maroon">D1</font>,<font color="Maroon">D2</font>;<br/><b>let </b><font color="Maroon">F1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="trees_3.html#M4">DTree-set</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">p1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F1</font>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> holds <br/>for <font color="Olive">T</font> being  <a href="trees_3.html#NM7">DecoratedTree</a> of <font color="Maroon">D1</font>,<font color="Maroon">D2</font>  st <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> <a href="trees_3.html#K10">`1</a> 
 ;<br/>

<b>set </b><font color="Maroon">W</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">W1</font> = <font color="Maroon">d1</font> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p1</font>;<br/>
<a NAME="E3:49"/><i><font color="Green">E47</font></i>: 
(  <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> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp;  <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">p1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> &amp;  <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> <font color="Maroon">p1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a>, <a class="ref" href="trees_3.html#T40">TREES_3:40</a></i>;<br/>
<a NAME="E4:49"/><b>then </b><i><font color="Green">E48</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> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p1</font></span>)</span> &amp;  <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="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><b>then reconsider </b><font color="Maroon">T</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="trees_3.html#M5">Element</a> of <font color="Maroon">F</font> <b>by </b><i/>;<br/><a NAME="E3:49_1"/>
<font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="trees_3.html#K10">`1</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:49_1"/><b>then </b>
( <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> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_3.html#K10">`1</a> </span>)</span> )
 <b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><b>hence </b><a NAME="E5:49_1"/>
<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> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:49"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K2">doms</a> <font color="Maroon">p1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_3.html#K10">`1</a> </span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <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/>
;<br/>
<b>hence </b><a NAME="E8:49"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_3.html#K10">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p1</font></span>)</span>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_4.html#D1" target="_self">TREES_4:def 1</a><br/>

<b>let </b><font color="Maroon">x</font> be   <a href="trees_4.html#NM1">Node</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_3.html#K10">`1</a> </span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="trees_4.html#NM1">Node</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E10:49"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_3.html#K10">`1</a> </span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="domain_1.html#K2">`1</a> 
 
<b>by </b><i><a class="ref" href="trees_3.html#T41">TREES_3:41</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_3_1"><b>suppose </b></a><a NAME="E1:49_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:49_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:49_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_3.html#K10">`1</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_3_2"><b>suppose </b></a><a NAME="E1:49_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><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">T</font> being   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">q</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:49_3_2"/><i><font color="Green">E54</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> &amp; <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <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="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as    <a href="trees_3.html#M5">Element</a> of <font color="Maroon">F</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="trees_4.html#NM1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_3.html#K10">`1</a> </span>)</span> <b>by </b><i/>;<br/><a NAME="E6:49_3_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:49_3_2"/><b>then </b>
<font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="trees_3.html#K10">`1</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E8:49_3_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_2.html#K3">.</a> <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">q</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_3.html#K10">`1</a> </span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">q</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E9:49_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="domain_1.html#K1">]</a></span> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_3.html#K10">`1</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="trees_4.html#K8" target="_self">-tree</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="trees_3.html#T41">TREES_3:41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
