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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">SCS</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="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>, <font color="Maroon">w</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>;<br/><b>let </b><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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>;<br/><b>let </b><font color="Maroon">e2</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>;<br/><b>let </b><font color="Maroon">q1</font> be  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">k1</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">k</font> = <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">eke</font> = <font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <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>,<font color="Maroon">e2</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E30</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="E2:14"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">e1</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">q1</font>
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font>
 <b>and </b><br/><a NAME="E4:14"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">q1</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#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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 ;<br/>

<a NAME="E5:14"/><i><font color="Green">E37</font></i>: 
 <a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">SCS</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">SCS</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">SCS</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/>
<a NAME="E6:14"/><b>then </b><i><font color="Green">E40</font></i>: 
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">SCS</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">SCS</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="E7:14"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a>  <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">SCS</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</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/>
<a NAME="E9:14"/><i><font color="Green">E42</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">SCS</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="E10:14"/>
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">nt</font> = <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> 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">SCS</font></span>)</span> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:14"/>
<font color="Maroon">e1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">nt</font>
 
<b>by </b><i>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">p'</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">SCS</font></span>)</span><b> such that </b><br/><a NAME="E14:14"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">p'</font>
 <b>and </b><br/><a NAME="E15:14"/>
<font color="Maroon">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">p'</font>
 <b>by </b><i>, <a class="ref" href="dtconstr.html#T10">DTCONSTR:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1'</font> = <font color="Maroon">q1</font> as    <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">SCS</font></span>)</span> <b>by </b><i>, , <a class="ref" href="trees_4.html#T15">TREES_4:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e1'</font> = <font color="Maroon">e1</font> as   <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/>
<b>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="E19:14"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 <b>and </b><br/><a NAME="E20:14"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">e1'</font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</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_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font> <b>by </b><i>, , , <a class="ref" href="pre_circ.html#T17">PRE_CIRC:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m</font> 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="E24:14"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">m'</font>,<font color="Maroon">e2</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">qq</font>
 <b>and </b><br/><a NAME="E25:14"/><i><font color="Green">E51</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">q1</font>
 <b>and </b><br/><a NAME="E26:14"/><i><font color="Green">E52</font></i>: 
<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">e2</font>
 <b>and </b><br/><a NAME="E27:14"/><i><font color="Green">E53</font></i>: 
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">q1</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">q1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><i>, <a class="ref" href="pre_circ.html#T19">PRE_CIRC:19</a></i>;<br/>
<a NAME="E28:14"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">qq</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">O</font> = <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> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3">coprod</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">R</font> =  <a href="msafree.html#K4">REL</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> as    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">O</font>,<font color="Maroon">O</font> <a href="finseq_1.html#K13">*</a>  ;<br/>
<a NAME="E31:14"/><i><font color="Green">E56</font></i>: 
 <a href="msafree.html#K5">DTConMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="hidden.html#R1">=</a>  <a href="lang1.html#G1">DTConstrStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">R</font> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D10">MSAFREE:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">TSDT</font> =  <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">SCS</font></span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5">FinTrees</a> <font color="Maroon">O</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_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="E1:14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TSDT</font>
 ;<br/><a NAME="E2:14_1"/><b>then </b>
<font color="Maroon">x</font> is    <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> <font color="Maroon">O</font>
 ;<br/><b>hence </b><a NAME="E3:14_1"/>
<font color="Maroon">x</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">O</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">TSDT</font> = <font color="Maroon">TSDT</font> as    <a href="trees_3.html#M4">DTree-set</a> of <font color="Maroon">O</font> <b>by </b><i><a class="ref" href="trees_3.html#D6">TREES_3:def 6</a></i>;<br/>
<a NAME="E35:14"/>
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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</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">SCS</font>,<font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E36:14"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">e2</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">SCS</font>,<font color="Maroon">w</font>
 
;<br/>
<a NAME="E37:14"/><b>then </b>
<font color="Maroon">e2</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">SCS</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">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</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">w</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">w</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">aa</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">SCS</font></span>)</span><b> such that </b><br/><a NAME="E39:14"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font>
 <b>and </b><br/><a NAME="E40:14"/><i><font color="Green">E61</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">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">aa</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">w</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">aa</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">w</font> ) )
 ;<br/>
<a NAME="E41:14"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">qq</font> <a href="tarski.html#R1">c=</a>  <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">SCS</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">qq</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_2"/><i><font color="Green">E62</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">qq</font>
 <b>and </b><br/><a NAME="E4:14_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qq</font> <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">x</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="ref" href="circuit1.html#D1" target="_self">Def1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <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">SCS</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:14_2_1_2"/><b>then </b>
<font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="circuit1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:14_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <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">SCS</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="circuit1.html#D1" target="_self">Def1</a>, , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">qq</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">TSDT</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rq</font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">q'</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">O</font> ;<br/>
<b>reconsider </b><font color="Maroon">rq</font> = <font color="Maroon">rq</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">O</font> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E45:14"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rq</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">qq</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E46:14"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">rq</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qq</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E47:14"/><i><font color="Green">E67</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>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E48:14"/><b>then </b><i><font color="Green">E68</font></i>: 
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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">av</font></span>)</span> <a href="hidden.html#R1">=</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</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="E49:14"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</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="ref" href="msafree2.html#T13">MSAFREE2:13</a></i>;<br/>
<a NAME="E50:14"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rq</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="circuit1.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E51:14"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rq</font> holds <br/>( ( <font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</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> implies for <font color="Olive">o1</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</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">rq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> holds <br/> <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</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="Olive">x</font> ) &amp; ( <font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3">coprod</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font></span>)</span> implies <font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K2">coprod</a> <span class="p1">(<span class="default"><span class="p2">(<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="Olive">x</font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>proof </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="E1:14_3"/><i><font color="Green">E71</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">rq</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:14_3"/><i><font color="Green">E73</font></i>: 
<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="finseq_4.html#K4">/.</a> <font color="Maroon">x'</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="txt" href="circuit1.html#E1:14"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">T</font> being   <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E5:14_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E6:14_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="circuit1.html#D1" target="_self">Def1</a>, , <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E7:14_3"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon">T'</font> being   <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E9:14_3"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">T'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E10:14_3"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T'</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="circuit1.html#D1" target="_self">Def1</a>, , <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1'</font> = <font color="Maroon">q1'</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">TSDT</font> ;<br/>
<b>reconsider </b><font color="Maroon">b</font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">q1'</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><span class="p2"><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="p3"><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> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="msafree.html#K3">coprod</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E13:14_3"/><i><font color="Green">E120</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</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="E14:14_3"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font> holds <br/><font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</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">SCS</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">av</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:14_3_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font>
 ;<br/>

<a NAME="E2:14_3_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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">SCS</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> <font color="Maroon">av</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="msafree2.html#T14">MSAFREE2:14</a></i>;<br/>
<a NAME="E3:14_3_1"/>
<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">n</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="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:14_3_1"/>
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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">SCS</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">av</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="circuit1.html#E6:14"><i><font color="Green">E40</font></i></a>, <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:14_3"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">q1'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">av</font>
 
<b>by </b><i>, <a class="ref" href="msafree.html#T9">MSAFREE:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="finseq_1.html#NM1">FinSequence</a> ;<br/>
<a NAME="E17:14_3"/>
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/>
<a NAME="E18:14_3"/><b>then </b>
<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="hidden.html#R2">in</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">av'</font> = <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> as   <a href="lang1.html#NM1">Symbol</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">SCS</font></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E20:14_3"/>
 <a href="msafree.html#K6">Sym</a> <font color="Maroon">av</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</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>
 
<b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>;<br/>
<a NAME="E21:14_3"/><b>then </b>
<font color="Maroon">av'</font> <a href="lang1.html#R1">==&gt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="msafree.html#T10">MSAFREE:10</a></i>;<br/>
<a NAME="E22:14_3"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">av'</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<a NAME="E23:14_3"/><i><font color="Green">E126</font></i>: 
<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="finseq_4.html#K4">/.</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">w</font>
 
<b>by </b><i>, , , , </i>;<br/>
<b>thus </b><a NAME="E24:14_3"/>
( <font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</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> implies for <font color="Olive">o1</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</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">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> holds <br/> <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">o1</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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_3_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</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>
 ;<br/>

<b>let </b><font color="Maroon">o1</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font>;<br/>

<b>assume </b><a NAME="E2:14_3_2"/><i><font color="Green">E128</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</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">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_3_2_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_2_1_1"><b>suppose </b></a><a NAME="E1:14_3_2_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_3_2_1_1_1_1"/>
(  ex <font color="Olive">xx</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">xx</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">aa</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">xx</font>,<font color="Maroon">w</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">aa</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">w</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:14_3_2_1_1_1_1_1"/>
 ex <font color="Olive">xx</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">xx</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">aa</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">xx</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_3_2_1_1_1_1_1"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E4:14_3_2_1_1_1_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">aa</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">xx</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E5:14_3_2_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</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> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xx</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="txt" href="circuit1.html#E3:14"><i><font color="Green">E33</font></i></a>, , , <a class="txt" href="circuit1.html#E14:14"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><a NAME="E6:14_3_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E7:14_3_2_1_1_1_1_1"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 ;<br/><b>hence </b><a NAME="E8:14_3_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="circuit1.html#E19:14"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:14_3_2_1_1_1_1_2"/>
 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">aa</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">w</font> )
 ;<br/><div class="add"><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="E3:14_3_2_1_1_1_1_2"/><i><font color="Green">E134</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">aa</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E4:14_3_2_1_1_1_1_2"/><i><font color="Green">E135</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">w</font>
 ;<br/><b>thus </b><a NAME="E5:14_3_2_1_1_1_1_2"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</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="txt" href="circuit1.html#E2:14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circuit1.html#E3:14"><i><font color="Green">E33</font></i></a>, , <a class="txt" href="circuit1.html#E7:14"><i><font color="Green">E41</font></i></a>, , <a class="txt" href="circuit1.html#E14:14"><i><font color="Green">E44</font></i></a>, , <a class="txt" href="circuit1.html#E19:14"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:14_3_2_1_1"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</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>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_2_1_2"><b>suppose </b></a><a NAME="E1:14_3_2_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:14_3_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</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="circuit1.html#D1" target="_self">Def1</a>, , <a class="txt" href="circuit1.html#E3:14"><i><font color="Green">E33</font></i></a>, , , <a class="txt" href="circuit1.html#E5:14"><i><font color="Green">E37</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:14_3_2_1_2"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</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="circuit1.html#D1" target="_self">Def1</a>, , <a class="txt" href="circuit1.html#E4:14"><i><font color="Green">E35</font></i></a>, <a class="txt" href="circuit1.html#E6:14"><i><font color="Green">E40</font></i></a>, <a class="txt" href="circuit1.html#E9:14"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="msafree1.html#T3">MSAFREE1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E25:14_3"/>
( <font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3">coprod</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font></span>)</span> implies <font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K2">coprod</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_3_3"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3">coprod</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font></span>)</span>
 ;<br/>

<a NAME="E2:14_3_3"/><b>then </b>
<font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</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">SCS</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree.html#T6">MSAFREE:6</a></i>;<br/>
<b>then consider </b><font color="Maroon">s1</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">IIG</font>, <font color="Maroon">x''</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:14_3_3"/><i><font color="Green">E139</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">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E5:14_3_3"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x''</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="msafree.html#T7">MSAFREE:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_3_3_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_3_1_1"><b>suppose </b></a><a NAME="E1:14_3_3_1_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:14_3_3_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">e2</font> <a href="hidden.html#R2">in</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">rqx</font> = <font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> as   <a href="dtconstr.html#NM2">Terminal</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">SCS</font></span>)</span> <b>by </b><i><a class="txt" href="circuit1.html#E9:14"><i><font color="Green">E42</font></i></a>, <a class="ref" href="msafree.html#T6">MSAFREE:6</a></i>;<br/><a NAME="E4:14_3_3_1_1"/>
<font color="Maroon">aa</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">rqx</font>
 <b>by </b><i>, , <a class="txt" href="circuit1.html#E3:14"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="dtconstr.html#T9">DTCONSTR:9</a></i>;<br/><a NAME="E5:14_3_3_1_1"/><b>then </b>
<font color="Maroon">aa</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">SCS</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">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</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">s1</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">s1</font> ) ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="circuit1.html#E14:14"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E6:14_3_3_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">aa</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">SCS</font>,<font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="msafree.html#D12">MSAFREE:def 12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_3_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:14_3_3_1_1_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">e2</font> <a href="hidden.html#R2">in</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 <b>by </b><i>, , <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/><b>assume </b><a NAME="E2:14_3_3_1_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s1</font>
 ;<br/><a NAME="E3:14_3_3_1_1_1"/><b>then </b>
<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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="subset_1.html#R1">misses</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="msafree.html#T13">MSAFREE:13</a></i>;<br/><a NAME="E4:14_3_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E5:14_3_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="circuit1.html#E19:14"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:14_3_3_1_1"/>
<font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K2">coprod</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font>
 <b>by </b><i><a class="txt" href="circuit1.html#E2:14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="circuit1.html#E7:14"><i><font color="Green">E41</font></i></a>, , <a class="txt" href="circuit1.html#E14:14"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="msafree.html#D2">MSAFREE:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_3_1_2"><b>suppose </b></a><a NAME="E1:14_3_3_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:14_3_3_1_2"/><b>then </b>
<font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">q1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , <a class="ref" href="circuit1.html#D1" target="_self">Def1</a>, , <a class="txt" href="circuit1.html#E3:14"><i><font color="Green">E33</font></i></a>, , , <a class="txt" href="circuit1.html#E5:14"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E3:14_3_3_1_2"/>
<font color="Maroon">rq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K2">coprod</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font>
 <b>by </b><i>, <a class="ref" href="circuit1.html#D1" target="_self">Def1</a>, , <a class="txt" href="circuit1.html#E4:14"><i><font color="Green">E35</font></i></a>, <a class="txt" href="circuit1.html#E6:14"><i><font color="Green">E40</font></i></a>, <a class="txt" href="circuit1.html#E9:14"><i><font color="Green">E42</font></i></a>, <a class="ref" href="msafree1.html#T3">MSAFREE1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E52:14"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">nt</font>,<span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">qq</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K4">REL</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font>
 
<b>by </b><i>, , , <a class="ref" href="msafree.html#T5">MSAFREE:5</a></i>;<br/>
<a NAME="E53:14"/><b>then </b>
<font color="Maroon">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <font color="Maroon">qq</font>
 
<b>by </b><i>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q'</font> as    <a href="dtconstr.html#M2">SubtreeSeq</a> of <font color="Maroon">nt</font> <b>by </b><i><a class="ref" href="dtconstr.html#D9">DTCONSTR:def 9</a></i>;<br/>
<a NAME="E55:14"/>
<font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <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>,<font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">nt</font> <a href="dtconstr.html#K9">-tree</a> <font color="Maroon">q'</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">eke'</font> = <font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <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>,<font color="Maroon">e2</font> as    <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">SCS</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">q''</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T29">FINSEQ_1:29</a></i>;<br/>
<a NAME="E58:14"/>
<font color="Maroon">q''</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">m'</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E59:14"/><b>then </b><i><font color="Green">E150</font></i>: 
( ( not <font color="Maroon">m'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q''</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">k1</font></span><a href="lang1.html#K3">*&gt;</a></span>,<font color="Maroon">e2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q''</font> ) or  ex <font color="Olive">r</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">q''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">k1</font></span><a href="lang1.html#K3">*&gt;</a></span>,<font color="Maroon">e2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 
<b>by </b><i><a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<a NAME="E60:14"/>
for <font color="Olive">r</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   holds not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T48">FINSEQ_1:48</a></i>;<br/>
<a NAME="E61:14"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">k1</font></span><a href="lang1.html#K3">*&gt;</a></span>,<font color="Maroon">e2</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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>
 
<b>by </b><i>, , <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_4"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">av</font> = <font color="Maroon">av</font>;<br/><a NAME="E1:14_4"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">av</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/><b>hence </b><a NAME="E2:14_4"/>
 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> <span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">k1</font></span><a href="lang1.html#K3">*&gt;</a></span>,<font color="Maroon">e2</font></span>)</span> <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><a class="txt" href="circuit1.html#E2:14"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E63:14"/><b>then </b>
<font color="Maroon">eke'</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">SCS</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">SCS</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> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">eke'</font> = <font color="Maroon">eke'</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <b>by </b><i>, <a class="ref" href="msafree.html#D12">MSAFREE:def 12</a></i>;<br/>
<a NAME="E65:14"/>
<font color="Maroon">eke'</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 
;<br/>
<b>hence </b><a NAME="E66:14"/>
<font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <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>,<font color="Maroon">e2</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 ;<br/>


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