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

<b>let </b><font color="Maroon">IIG</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a> <a href="msafree2.html#V5">monotonic</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">IIG</font>;<br/><b>let </b><font color="Maroon">iv</font> be    <a href="msafree2.html#M1">InputValues</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>, <font color="Maroon">e1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">t1</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>











<b>set </b><font color="Maroon">X</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">FX</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span>;<br/>
<a NAME="E1:9"/><i><font color="Green">E42</font></i>: 
 <a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">v</font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font><br/> for <font color="Olive">e</font>, <font color="Olive">e1</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font><br/> for <font color="Olive">t</font>, <font color="Olive">t1</font> being  <a href="trees_2.html#NM4">DecoratedTree</a>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> &amp; <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Olive">e1</font> &amp;  <a href="circuit1.html#K6">depth</a> <font color="Olive">v</font>,<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">e1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Olive">v</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font> holds <br/> <a href="relat_1.html#K1">dom</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">t1</font>;<br/>
<a NAME="E2:9"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>, <font color="Maroon">e1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">t1</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:9_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> )
 <b>and </b><br/><a NAME="E2:9_1"/><i><font color="Green">E45</font></i>: 
 <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>and </b><br/><a NAME="E3:9_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font>
 ;<br/>

<a NAME="E4:9_1"/><i><font color="Green">E48</font></i>: 
 <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E5:9_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 
;<br/>
<a NAME="E6:9_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,<font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E7:9_1"/>
 <a href="msafree.html#K7">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> : (  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="msafree.html#D12">MSAFREE:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E9:9_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E10:9_1"/><i><font color="Green">E54</font></i>: 
(  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) )
 <b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or ( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> ) ) or ( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> &amp;  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) )
 </span><b>by </b><i>, , <a class="ref" href="circuit1.html#T19">CIRCUIT1:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E4:9_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="msafree2.html#T2">MSAFREE2:2</a></i>;<br/><a NAME="E5:9_1_1_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">iv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , , </i>;<br/><b>thus </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font>
<b>by </b><i>, , , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:9_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 <b>and </b><br/><a NAME="E2:9_1_1_2"/><i><font color="Green">E65</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:9_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E5:9_1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:9_1_1_2"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , </i>;<br/><b>thus </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font>
<b>by </b><i>, , , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:9_1_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 <b>and </b><br/><a NAME="E2:9_1_1_3"/><i><font color="Green">E69</font></i>: 
 ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E4:9_1_1_3"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E5:9_1_1_3"/><i><font color="Green">E73</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><a NAME="E6:9_1_1_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">IIG</font> : <font color="Olive">s</font> is <a href="msualg_2.html#V1">with_const_op</a> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="msafree2.html#D1">MSAFREE2:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E8:9_1_1_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E9:9_1_1_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">s</font> is <a href="msualg_2.html#V1">with_const_op</a>
 ;<br/><b>consider </b><font color="Maroon">o'</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E11:9_1_1_3"/><i><font color="Green">E76</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E12:9_1_1_3"/><i><font color="Green">E78</font></i>: 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="msualg_2.html#D2">MSUALG_2:def 2</a></i>;<br/><a NAME="E13:9_1_1_3"/><i><font color="Green">E80</font></i>: 
 <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font>
 <b>by </b><i><a class="ref" href="msafree2.html#T5">MSAFREE2:5</a></i>;<br/><a NAME="E14:9_1_1_3"/><i><font color="Green">E81</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/><a NAME="E15:9_1_1_3"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">o'</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/><a NAME="E16:9_1_1_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font>
 <b>by </b><i>, , , <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E18:9_1_1_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="circuit1.html#T10">CIRCUIT1:10</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o'</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="msafree2.html#T13">MSAFREE2:13</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>
;<br/><a NAME="E20:9_1_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E21:9_1_1_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="trees_4.html#T20">TREES_4:20</a></i>;<br/><a NAME="E22:9_1_1_3"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , </i>;<br/><b>thus </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , <a class="txt" href="circuit2.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font>
<b>by </b><i>, , , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:9"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:9_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>, <font color="Maroon">e1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">t1</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>





<b>assume </b><a NAME="E2:9_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp;  <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> )
 ;<br/>

<a NAME="E3:9_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 
;<br/>
<a NAME="E4:9_2"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,<font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E5:9_2"/>
 <a href="msafree.html#K7">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> : (  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="msafree.html#D12">MSAFREE:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E7:9_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E8:9_2"/><i><font color="Green">E94</font></i>: 
(  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E9:9_2"/>
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E10:9_2"/><b>then </b><i><font color="Green">E95</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:9_2"/>
 <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font>
 
<b>by </b><i><a class="ref" href="msafree2.html#T5">MSAFREE2:5</a></i>;<br/>
<a NAME="E12:9_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_2_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or ( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font></span>)</span> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> ) ) or ( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> &amp;  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> ) ) or ( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font> &amp;  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) )
 </span><b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_2_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E4:9_2_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="msafree2.html#T2">MSAFREE2:2</a></i>;<br/><a NAME="E5:9_2_1_1"/><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">iv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , , </i>;<br/><b>thus </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font>
<b>by </b><i>, , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font></span>)</span>
 <b>and </b><br/><a NAME="E2:9_2_1_2"/><i><font color="Green">E115</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:9_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E5:9_2_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:9_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 <b>and </b><br/><a NAME="E2:9_2_1_3"/><i><font color="Green">E140</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:9_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E5:9_2_1_3"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:9_2_1_3"/><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , </i>;<br/><b>thus </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font>
<b>by </b><i>, , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_4"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font>
 <b>and </b><br/><a NAME="E2:9_2_1_4"/><i><font color="Green">E144</font></i>: 
 ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E4:9_2_1_4"/><i><font color="Green">E145</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E5:9_2_1_4"/><i><font color="Green">E146</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><a NAME="E6:9_2_1_4"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:9_2_1_4"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="circuit1.html#T10">CIRCUIT1:10</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p4">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E10:9_2_1_4"/><i><font color="Green">E150</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E11:9_2_1_4"/><i><font color="Green">E151</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i><a NAME="E12:9_2_1_4"/><i><font color="Green">E152</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:9_2_1_4"/>
 <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="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="msafree2.html#T13">MSAFREE2:13</a></i>;<br/><a NAME="E14:9_2_1_4"/><b>then </b><i><font color="Green">E153</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E15:9_2_1_4"/><i><font color="Green">E154</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="E16:9_2_1_4"/><i><font color="Green">E155</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</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="9_2_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_2_1_4_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>set </b><font color="Maroon">v1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>;<br/><a NAME="E3:9_2_1_4_1"/><i><font color="Green">E158</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>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E4:9_2_1_4_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="txt" href="circuit2.html#E1"><i><font color="Green">Lemma32</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">ee</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i>, , , , <a class="txt" href="circuit2.html#E1"><i><font color="Green">Lemma32</font></i></a>, , <a class="ref" href="msafree2.html#T14">MSAFREE2:14</a></i>;<br/><b>reconsider </b><font color="Maroon">fv1</font> = <span class="p1">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> ;<br/><a NAME="E7:9_2_1_4_1"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fv1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ee</font>
 <b>by </b><i>, , <a class="ref" href="circuit2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E8:9_2_1_4_1"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="trees_3.html#T26">TREES_3:26</a></i>;<br/><a NAME="E19:9_2_1_4"/><i><font color="Green">E161</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font>
 <b>by </b><i>, , , , <a class="txt" href="circuit2.html#E1"><i><font color="Green">Lemma32</font></i></a>, </i>;<br/><a NAME="E20:9_2_1_4"/><i><font color="Green">E162</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <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">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">v1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>;<br/><b>assume </b><a NAME="E1:9_2_1_4_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">i</font> <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>
 ;<br/><a NAME="E2:9_2_1_4_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p4">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:9_2_1_4_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">ee</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i>, , , , , , <a class="ref" href="msafree2.html#T14">MSAFREE2:14</a></i>;<br/><b>reconsider </b><font color="Maroon">fv1</font> = <span class="p1">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> ;<br/><a NAME="E6:9_2_1_4_2"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fv1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ee</font>
 <b>by </b><i>, , , </i>;<br/><b>then reconsider </b><font color="Maroon">ee1</font> = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> ;<br/><a NAME="E8:9_2_1_4_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E9:9_2_1_4_2"/><b>then </b>
 <a href="circuit1.html#K6">depth</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="circuit1.html#T20">CIRCUIT1:20</a></i>;<br/><a NAME="E10:9_2_1_4_2"/><b>then </b>
 <a href="circuit1.html#K6">depth</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:9_2_1_4_2"/><b>then </b>
 <a href="circuit1.html#K6">depth</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:9_2_1_4_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ee</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ee1</font>
 <b>by </b><i>, </i>;<br/><b>hence </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="relat_1.html#K1">dom</a> <font color="Maroon">ee1</font>
<b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="txt" href="circuit2.html#E1"><i><font color="Green">Lemma32</font></i></a>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E22:9_2_1_4"/><b>then </b>
 <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">q</font>
 <b>by </b><i>, <a class="txt" href="circuit2.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="circuit2.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/><b>hence </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> = 
 <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">q</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="trees_4.html#T10">TREES_4:10</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font>
<b>by </b><i>, , <a class="ref" href="trees_4.html#T10">TREES_4:10</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:9"/><i><font color="Green">E166</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>reconsider </b><font color="Maroon">k</font> =  <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> as    <a href="subset_1.html#M2">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="E6:9"/>
 <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
;<br/>
<b>hence </b><a NAME="E7:9"/>
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">iv</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> implies  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font> )
 <b>by </b><i/>;<br/>


</div>
