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

<b>let </b><font color="Maroon">IIG</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a> <a href="msafree2.html#V5">monotonic</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">IIG</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>;<br/>





<b>consider </b><font color="Maroon">s</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:31"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" target="_self">depth</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E32</font></i>: 
 <a href="circuit1.html#K6" target="_self">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">s</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V2">real-membered</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E5:31"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" target="_self">depth</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : verum </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E7:31"/><i><font color="Green">E33</font></i>: 
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y</font>
 ;<br/>
<b>consider </b><font color="Maroon">t2</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E9:31"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 <b>and </b><br/><a NAME="E10:31"/><i><font color="Green">E37</font></i>: 
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t2</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">dt</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">t'</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E12:31"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 <b>and </b><br/><a NAME="E13:31"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dt</font>
 <b>and </b><br/><a NAME="E14:31"/><i><font color="Green">E42</font></i>: 
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">t'</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:31"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t'</font>
 <b>and </b><br/><a NAME="E17:31"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">t'</font>
 <b>by </b><i><a class="ref" href="trees_1.html#D15">TREES_1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon">ss</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="E19:31"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">ss</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">tt</font></span>)</span> where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E20:31"/><i><font color="Green">E50</font></i>: 
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">ss</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">YY</font> = <font color="Maroon">ss</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">t''</font> = <font color="Maroon">t</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>thus </b><a NAME="E23:31"/>
(  not  <a href="circuit1.html#K6" target="_self">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E51</font></i>: 
 <a href="circuit1.html#K6" target="_self">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , , , <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_1.html#T80">TREES_1:80</a></i>;<br/>
<a NAME="E3:31_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">t''</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T14">FUNCT_1:14</a></i>;<br/>
<a NAME="E4:31_1"/><b>then </b>
<font color="Maroon">t''</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , , , <a class="ref" href="pre_circ.html#T2">PRE_CIRC:2</a></i>;<br/>
<a NAME="E5:31_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/>
<a NAME="E6:31_1"/><b>then </b><i><font color="Green">E53</font></i>: 
1 <a href="hidden.html#R2">in</a> <font color="Maroon">YY</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:31_1"/>
for <font color="Olive">kk</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">kk</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YY</font> holds <br/><font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">kk</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:31_1_1"/>
<font color="Maroon">kk</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YY</font>
 ;<br/>

<b>then consider </b><font color="Maroon">tt</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">tt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:31_1_1"/>
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">tt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:31_1_1"/><b>then </b>
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">tt</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, , <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E6:31_1_1"/><b>then </b>
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">tt</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E7:31_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">tt</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>consider </b><font color="Maroon">tiv</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E9:31_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">tt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tiv</font>
 <b>and </b><br/><a NAME="E10:31_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="circuit1.html#K5" target="_self">depth</a> <font color="Maroon">tt</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K9">depth</a> <font color="Maroon">tiv</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">dt'</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">t'''</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E12:31_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">dt'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tiv</font>
 <b>and </b><br/><a NAME="E13:31_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">t'''</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dt'</font>
 <b>and </b><br/><a NAME="E14:31_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">tiv</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">t'''</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>
<a NAME="E15:31_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">t'''</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E3:31"><i><font color="Green">E32</font></i></a>, , <a class="txt" href="circuit1.html#E10:31"><i><font color="Green">E37</font></i></a>, <a class="ref" href="trees_1.html#T56">TREES_1:56</a>, <a class="ref" href="trees_1.html#T80">TREES_1:80</a></i>;<br/>
<a NAME="E16:31_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E7:31"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit1.html#E9:31"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="funct_1.html#T14">FUNCT_1:14</a></i>;<br/>
<a NAME="E17:31_1_1"/><b>then </b>
<font color="Maroon">tt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p3">(<span class="default"><font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E7:31"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit1.html#E9:31"><i><font color="Green">E35</font></i></a>, , , <a class="ref" href="pre_circ.html#T2">PRE_CIRC:2</a></i>;<br/>
<b>hence </b><a NAME="E18:31_1_1"/>
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:31_1"/><b>then </b>
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="txt" href="circuit1.html#E2:31"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E9:31_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:31_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E24:31"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> )
 ;<br/>

<a NAME="E25:31"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E26:31"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E27:31"/><i><font color="Green">E67</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ss</font>
 
<b>by </b><i/>;<br/>
<a NAME="E28:31"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">ct</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">t</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E30:31"/>
<font color="Maroon">ct</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, , , <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E31:31"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or  <a href="card_1.html#K4">card</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><a NAME="E2:31_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E3:31_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>hence </b><a NAME="E4:31_2"/>
contradiction
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E34:31"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="circuit1.html#E2:31"><i><font color="Green">E30</font></i></a>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<a NAME="E35:31"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E37:31"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">t''</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E38:31"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="txt" href="circuit1.html#E3:31"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y''</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E40:31"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y''</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">t''</font>
 <b>by </b><i>, , , , <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E41:31"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y''</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"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="txt" href="circuit1.html#E7:31"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E42:31"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E43:31"/>
 <a href="circuit1.html#K6" target="_self">depth</a> <font color="Maroon">v</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , , , <a class="ref" href="circuit1.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>


</div>
