<?xml version="1.0"?>
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<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="msafree2.html#V4">locally-finite</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">IIG</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">IIG</font>, <font color="Maroon">v1</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">IIG</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E30</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="E2:32"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">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>
 ;<br/>

<b>consider </b><font color="Maroon">s</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" target="_self">depth</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">t</font> is    <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> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E5:32"/><i><font color="Green">E35</font></i>: 
 <a href="circuit1.html#K6" target="_self">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">s</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V2">real-membered</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E7:32"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" target="_self">depth</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">t</font> is    <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> : verum </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">t</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="Maroon">v</font><b> such that </b><br/><a NAME="E9:32"/><i><font color="Green">E37</font></i>: 
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y</font>
 ;<br/>
<b>consider </b><font color="Maroon">s1</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:32"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" target="_self">depth</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <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">v1</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E12:32"/><i><font color="Green">E41</font></i>: 
 <a href="circuit1.html#K6" target="_self">depth</a> <font color="Maroon">v1</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">s1</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Y1</font> = <font color="Maroon">s1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V2">real-membered</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E14:32"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" target="_self">depth</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">t1</font> is    <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">v1</font> : verum </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">t1</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="Maroon">v1</font><b> such that </b><br/><a NAME="E16:32"/><i><font color="Green">E42</font></i>: 
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y1</font>
 ;<br/>
<b>consider </b><font color="Maroon">e'</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="msafree.html#K11">FreeMSA</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">v1</font><b> such that </b><br/><a NAME="E18:32"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font>
 <b>and </b><br/><a NAME="E19:32"/><i><font color="Green">E46</font></i>: 
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K9">depth</a> <font color="Maroon">e'</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">dt1'</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">t1'</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E21:32"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">dt1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font>
 <b>and </b><br/><a NAME="E22:32"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">t1'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dt1'</font>
 <b>and </b><br/><a NAME="E23:32"/><i><font color="Green">E51</font></i>: 
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">e'</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">t1'</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>
<a NAME="E24:32"/><i><font color="Green">E52</font></i>: 
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">v1</font>,<font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">v</font>,<font color="Maroon">A</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E25:32"/>
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">v1</font>,<font color="Maroon">A</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E26:32"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">v1</font>,<font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E27:32"/><b>then </b>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <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#K1">SortsWithConstants</a> <font color="Maroon">IIG</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E28:32"/><b>then </b><i><font color="Green">E53</font></i>: 
( not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> &amp; not <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>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E29:32"/><b>then </b><i><font color="Green">E55</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>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E30:32"/>
 <a href="msafree2.html#K6">FreeEnv</a> <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/>
<a NAME="E31:32"/><b>then </b>
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> <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="E32:32"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</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>
 
;<br/>
<a NAME="E33:32"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</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="E35:32"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E36:32"/><i><font color="Green">E58</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> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:32_1"/>
<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="E2:32_1"/><i><font color="Green">E59</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>
 ;<br/><b>consider </b><font color="Maroon">e'</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="msafree.html#K11">FreeMSA</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><b> such that </b><br/><a NAME="E4:32_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font>
 <b>and </b><br/><a NAME="E5:32_1"/><i><font color="Green">E62</font></i>: 
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K9">depth</a> <font color="Maroon">e'</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">dta</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">ta</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E7:32_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">dta</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font>
 <b>and </b><br/><a NAME="E8:32_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">ta</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dta</font>
 <b>and </b><br/><a NAME="E9:32_1"/><i><font color="Green">E66</font></i>: 
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">e'</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">ta</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/><a NAME="E10:32_1"/>
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="circuit1.html#E2:32"><i><font color="Green">E32</font></i></a>, , <a class="txt" href="circuit1.html#E5:32"><i><font color="Green">E35</font></i></a>, , <a class="txt" href="circuit1.html#E9:32"><i><font color="Green">E37</font></i></a>, , , <a class="ref" href="trees_1.html#T79">TREES_1:79</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><b>hence </b><a NAME="E11:32_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="circuit1.html#E1:32"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>then 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="E39:32"/><i><font color="Green">E67</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="E40:32"/><i><font color="Green">E68</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="txt" href="circuit1.html#E4:32"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E41:32"/><i><font color="Green">E69</font></i>: 
 <a href="dtconstr.html#K7">NonTerminals</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">IIG</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="msafree.html#T6">MSAFREE:6</a></i>;<br/>
<a NAME="E42:32"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o'</font> = <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> as   <a href="dtconstr.html#NM3">NonTerminal</a> of <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>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="trees_4.html#M1">FinSequence</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="E45:32"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E46:32"/>
<font color="Maroon">o'</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="dtconstr.html#T10">DTCONSTR:10</a></i>;<br/>
<b>consider </b><font color="Maroon">q'</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="E48:32"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font>
 <b>and </b><br/><a NAME="E49:32"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">a</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">q'</font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit1.html#E9:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">av</font> =  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> ;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E52:32"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">av</font></span>)</span>
 <b>and </b><br/><a NAME="E53:32"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">v1</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">av</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="circuit1.html#E11:32"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E55:32"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">av</font>
 
<b>by </b><i>, <a class="txt" href="circuit1.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/>
<a NAME="E56:32"/><b>then </b><i><font color="Green">E118</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> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">av</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E2:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circuit1.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="circuit1.html#E9:32"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="msafree2.html#T13">MSAFREE2:13</a></i>;<br/>
<a NAME="E57:32"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">av</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E58:32"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q'</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E11:32"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="circuit1.html#E11:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_3.html#T28">FINSEQ_3:28</a></i>;<br/>
<a NAME="E60:32"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
;<br/>
<a NAME="E61:32"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">k1</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</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">q'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E11:32"><i><font color="Green">E40</font></i></a>, <a class="txt" href="circuit1.html#E18:32"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pre_circ.html#T17">PRE_CIRC:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">k1</font></span><a href="lang1.html#K3">*&gt;</a></span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>consider </b><font color="Maroon">qq</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="E64:32"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">t</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">f</font>,<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">qq</font>
 <b>and </b><br/><a NAME="E65:32"/><i><font color="Green">E125</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">qq</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q'</font>
 <b>and </b><br/><a NAME="E66:32"/>
<font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font>
 <b>and </b><br/><a NAME="E67:32"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">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">q'</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><i><a class="txt" href="circuit1.html#E2:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circuit1.html#E9:32"><i><font color="Green">E37</font></i></a>, , , , <a class="ref" href="pre_circ.html#T19">PRE_CIRC:19</a></i>;<br/>
<a NAME="E68:32"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">q'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">v1</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E2:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circuit1.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="circuit1.html#E9:32"><i><font color="Green">E37</font></i></a>, , <a class="txt" href="circuit1.html#E11:32"><i><font color="Green">E40</font></i></a>, <a class="txt" href="circuit1.html#E12:32"><i><font color="Green">E41</font></i></a>, <a class="txt" href="circuit1.html#E16:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="msafree2.html#T14">MSAFREE2:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">tft</font> = <font color="Maroon">t</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">f</font>,<font color="Maroon">t1</font> as   <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/>
<a NAME="E70:32"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">av</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="trees_4.html#K4">-tree</a> <font color="Maroon">q'</font>
 
<b>by </b><i>, <a class="txt" href="circuit1.html#E2:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circuit1.html#E5:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="circuit1.html#E9:32"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">tft</font> = <font color="Maroon">tft</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> <font color="Maroon">v</font> <b>by </b><i>, , <a class="txt" href="circuit1.html#E19:32"><i><font color="Green">E46</font></i></a>, <a class="txt" href="circuit1.html#E22:32"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">dtft</font> =  <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">tft</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E73:32"/>
<font color="Maroon">dtft</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E74:32"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">dtft</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">e'</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="msafree.html#K11">FreeMSA</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><b> such that </b><br/><a NAME="E76:32"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">tft</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font>
 <b>and </b><br/><a NAME="E77:32"/><i><font color="Green">E129</font></i>: 
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">tft</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K9">depth</a> <font color="Maroon">e'</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">dttft</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">ttft</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E79:32"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">dttft</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font>
 <b>and </b><br/><a NAME="E80:32"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">ttft</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dttft</font>
 <b>and </b><br/><a NAME="E81:32"/><i><font color="Green">E134</font></i>: 
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">e'</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">ttft</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>
<a NAME="E82:32"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">qq</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E11:32"><i><font color="Green">E40</font></i></a>, , <a class="txt" href="circuit1.html#E21:32"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>consider </b><font color="Maroon">qq'</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="E84:32"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">qq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qq'</font>
 <b>and </b><br/><a NAME="E85:32"/><i><font color="Green">E139</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">tft</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">qq'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">ttft</font> <b>by </b><i><a class="txt" href="circuit1.html#E19:32"><i><font color="Green">E46</font></i></a>, <a class="txt" href="circuit1.html#E24:32"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="circuit1.html#E29:32"><i><font color="Green">E55</font></i></a>, , <a class="txt" href="circuit1.html#E36:32"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="pre_circ.html#T17">PRE_CIRC:17</a></i>;<br/>
<a NAME="E87:32"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E88:32"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E2:32"><i><font color="Green">E32</font></i></a>, , , <a class="txt" href="circuit1.html#E19:32"><i><font color="Green">E46</font></i></a>, <a class="ref" href="pre_circ.html#T17">PRE_CIRC:17</a></i>;<br/>
<a NAME="E89:32"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">tft</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E2:32"><i><font color="Green">E32</font></i></a>, , , <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E90:32"/><b>then </b>
<font color="Maroon">ttft</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1'</font>
 
<b>by </b><i>, , , <a class="txt" href="circuit1.html#E24:32"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="circuit1.html#E29:32"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="trees_1.html#T66">TREES_1:66</a></i>;<br/>
<a NAME="E91:32"/><b>then </b>
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">tft</font>
 
<b>by </b><i><a class="ref" href="circuit1.html#D1" target="_self">Def1</a>, , <a class="txt" href="circuit1.html#E28:32"><i><font color="Green">E53</font></i></a>, <a class="txt" href="circuit1.html#E35:32"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="trees_1.html#T85">TREES_1:85</a></i>;<br/>
<b>hence </b><a NAME="E92:32"/>
 <a href="circuit1.html#K6" target="_self">depth</a> <font color="Maroon">v1</font>,<font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circuit1.html#K6" target="_self">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font>
 <b>by </b><i>, , , , <a class="txt" href="circuit1.html#E23:32"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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