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

<b>let </b><font color="Maroon">IIG</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a> <a href="msafree2.html#V5">monotonic</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">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>assume </b><b>that </b><br/><a NAME="E1:21"/><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="E2:21"/><i><font color="Green">E32</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">v</font>,<font color="Maroon">SCS</font>
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green">E33</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="E4:21"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">e2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font>
 ;<br/>

<b>assume </b><a NAME="E5:21"/><i><font color="Green">E37</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">e2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">w</font>,<font color="Maroon">SCS</font>
 ;<br/>

<b>consider </b><font color="Maroon">sw</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="E7:21"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">sw</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E8:21"/><i><font color="Green">E41</font></i>: 
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">w</font>,<font color="Maroon">SCS</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">sw</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">sw</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="E10:21"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">e2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:21"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">e2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E12:21"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">e2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E14:21"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</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">e3</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font><b> such that </b><br/><a NAME="E16:21"/><i><font color="Green">E44</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">e3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>
<b>consider </b><font color="Maroon">sv</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="E18:21"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">sv</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</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">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E19:21"/><i><font color="Green">E49</font></i>: 
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">v</font>,<font color="Maroon">SCS</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">sv</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">sv</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/>
<b>reconsider </b><font color="Maroon">q1'</font> = <font color="Maroon">q1</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:21"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q1'</font>
 <b>and </b><br/><a NAME="E24:21"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E25:21"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q1</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">kN</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">kN</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="ref" href="finseq_3.html#T28">FINSEQ_3:28</a></i>;<br/>
<a NAME="E28:21"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">kN</font>
 
;<br/>
<a NAME="E29:21"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">kN</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E30:21"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">kN</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E31:21"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</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="E34:21"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 <b>and </b><br/><a NAME="E35:21"/><i><font color="Green">E58</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">k'</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="circuit1.html#D1" target="_self">Def1</a>, , <a class="ref" href="pre_circ.html#T17">PRE_CIRC:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e3'</font> = <font color="Maroon">e3</font> as   <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/>
<b>reconsider </b><font color="Maroon">k''</font> = <font color="Maroon">k'</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">eke</font> = <font color="Maroon">e1'</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">k''</font>,<font color="Maroon">e3'</font> as   <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/>
<b>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>, , , , , </i>;<br/>
<a NAME="E41:21"/><i><font color="Green">E59</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">eke</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E42:21"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">k'</font>,<font color="Maroon">e3</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">k'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">e1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">e3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_circ.html#T23">PRE_CIRC:23</a></i>;<br/>
<a NAME="E43:21"/>
<font color="Maroon">e1</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font>
 
<b>by </b><i>, , , , <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E44:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">e1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">k'</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">e1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">e3</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E45:21"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">e1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">k'</font>,<font color="Maroon">e3</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E46:21"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>


</div>
