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<b>let </b><font color="Maroon" title="c1">IIG</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a> <a href="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</font>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>;<br/>





<b>consider </b><font color="Maroon" title="c4">s</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:31"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c4">s</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D6" target="_self" title="CIRCUIT1:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c4">s</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E5:31"/>
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="circuit1.html#E2:31"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font><b> such that </b><br/><a NAME="E7:31"/><i><font color="Green" title="E18">A3</font></i>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c5">Y</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">t2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font><b> such that </b><br/><a NAME="E9:31"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t2</font>
 <b>and </b><br/><a NAME="E10:31"/><i><font color="Green" title="E20">A5</font></i>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c7">t2</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D5" target="_self" title="CIRCUIT1:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">dt</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>, <font color="Maroon" title="c9">t'</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a><b> such that </b><br/><a NAME="E12:31"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c8">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t2</font>
 <b>and </b><br/><a NAME="E13:31"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c9">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">dt</font>
 <b>and </b><br/><a NAME="E14:31"/><i><font color="Green" title="E23">A8</font></i>: 
 <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c7">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c9">t'</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">p</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E16:31"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">t'</font>
 <b>and </b><br/><a NAME="E17:31"/><i><font color="Green" title="E25">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c9">t'</font>
 <b>by </b><i><a class="ref" href="trees_1.html#D15" title="TREES_1:def.15">TREES_1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">ss</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:31"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c11">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">tt</font></span>)</span> where <font color="Olive" title="b1">tt</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E20:31"/><i><font color="Green" title="E27">A12</font></i>: 
 <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c11">ss</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D4" target="_self" title="CIRCUIT1:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">YY</font> = <font color="Maroon" title="c11">ss</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">t''</font> = <font color="Maroon" title="c6">t</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>thus </b><a NAME="E23:31"/>
(  not  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> or <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">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" title="E28">A13</font></i>: 
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c9">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E3:31"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="circuit1.html#E7:31"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="circuit1.html#E10:31"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="circuit1.html#E14:31"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a>, <a class="ref" href="trees_1.html#T80" title="TREES_1:th.80">TREES_1:80</a></i>;<br/>
<a NAME="E3:31_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">t''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E9:31"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="circuit1.html#E12:31"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="circuit1.html#E13:31"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="funct_1.html#T14" title="FUNCT_1:th.14">FUNCT_1:14</a></i>;<br/>
<a NAME="E4:31_1"/><b>then </b>
<font color="Maroon" title="c13">t''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E9:31"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="circuit1.html#E12:31"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="circuit1.html#E13:31"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="circuit1.html#E2:31_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="pre_circ.html#T2" title="PRE_CIRC:th.2">PRE_CIRC:2</a></i>;<br/>
<a NAME="E5:31_1"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>;<br/>
<a NAME="E6:31_1"/><b>then </b><i><font color="Green" title="E30">A15</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">YY</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E19:31"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/>
<a NAME="E7:31_1"/>
for <font color="Olive" title="b1">kk</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">YY</font> holds <br/><font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="c14">kk</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c14">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">YY</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">tt</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font><b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green" title="E31">A16</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c15">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">kk</font>
 <b>by </b><i><a class="txt" href="circuit1.html#E19:31"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/>
<a NAME="E4:31_1_1"/>
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c15">tt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E2:31"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>
<a NAME="E5:31_1_1"/><b>then </b>
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c15">tt</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="circuit1.html#E3:31"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="circuit1.html#E1:31_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E6:31_1_1"/><b>then </b><i><font color="Green" title="E32">A17</font></i>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c15">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<b>consider </b><font color="Maroon" title="c16">tiv</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font><b> such that </b><br/><a NAME="E8:31_1_1"/><i><font color="Green" title="E33">A18</font></i>: 
<font color="Maroon" title="c15">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">tiv</font>
 <b>and </b><br/><a NAME="E9:31_1_1"/><i><font color="Green" title="E34">A19</font></i>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c15">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c16">tiv</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D5" target="_self" title="CIRCUIT1:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">dt'</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>, <font color="Maroon" title="c18">t'''</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a><b> such that </b><br/><a NAME="E11:31_1_1"/><i><font color="Green" title="E35">A20</font></i>: 
<font color="Maroon" title="c17">dt'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">tiv</font>
 <b>and </b><br/><a NAME="E12:31_1_1"/><i><font color="Green" title="E36">A21</font></i>: 
<font color="Maroon" title="c18">t'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">dt'</font>
 <b>and </b><br/><a NAME="E13:31_1_1"/><i><font color="Green" title="E37">A22</font></i>: 
 <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c16">tiv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c18">t'''</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></i>;<br/>
<a NAME="E14:31_1_1"/><i><font color="Green" title="E38">A23</font></i>: 
<font color="Maroon" title="c18">t'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E6:31_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="circuit1.html#E9:31_1_1"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="circuit1.html#E13:31_1_1"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a>, <a class="ref" href="trees_1.html#T80" title="TREES_1:th.80">TREES_1:80</a></i>;<br/>
<a NAME="E15:31_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E8:31_1_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="circuit1.html#E11:31_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="circuit1.html#E12:31_1_1"><i><font color="Green" title="E36">A21</font></i></a>, <a class="ref" href="funct_1.html#T14" title="FUNCT_1:th.14">FUNCT_1:14</a></i>;<br/>
<a NAME="E16:31_1_1"/><b>then </b>
<font color="Maroon" title="c15">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c15">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E8:31_1_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="circuit1.html#E11:31_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="circuit1.html#E12:31_1_1"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="circuit1.html#E14:31_1_1"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="pre_circ.html#T2" title="PRE_CIRC:th.2">PRE_CIRC:2</a></i>;<br/>
<b>hence </b><a NAME="E17:31_1_1"/>
<font color="Maroon" title="c14">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="circuit1.html#E3:31_1_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:31_1"/><b>then </b>
 <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="circuit1.html#E20:31"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="circuit1.html#E6:31_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E9:31_1"/><b>then </b>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="circuit1.html#T11" target="_self" title="CIRCUIT1:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E10:31_1"/>
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> or <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E24:31"/>
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> or <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font> )
 ;<br/>

<a NAME="E25:31"/><b>then </b>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E26:31"/><b>then </b><i><font color="Green" title="E28">A24</font></i>: 
 <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="circuit1.html#T11" target="_self" title="CIRCUIT1:th.11">Th11</a></i>;<br/>
<a NAME="E27:31"/><i><font color="Green" title="E29">A25</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ss</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E19:31"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/>
<a NAME="E28:31"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">ct</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">t</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E30:31"/>
<font color="Maroon" title="c14">ct</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="circuit1.html#E20:31"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="circuit1.html#E26:31"><i><font color="Green" title="E28">A24</font></i></a>, <a class="txt" href="circuit1.html#E27:31"><i><font color="Green" title="E29">A25</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E31:31"/><b>then </b><i><font color="Green" title="E30">A26</font></i>: 
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 or  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</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" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><a NAME="E2:31_2"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><b>hence </b><a NAME="E3:31_2"/>
contradiction
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E34:31"/><i><font color="Green" title="E31">A27</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="circuit1.html#E31:31"><i><font color="Green" title="E30">A26</font></i></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/>
<a NAME="E35:31"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">t</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E37:31"/><i><font color="Green" title="E32">A28</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c16">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">t''</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E38:31"/><i><font color="Green" title="E33">A29</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c16">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E34:31"><i><font color="Green" title="E31">A27</font></i></a>, <a class="txt" href="circuit1.html#E37:31"><i><font color="Green" title="E32">A28</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">y''</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E40:31"/><i><font color="Green" title="E34">A30</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c17">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">t''</font>
 <b>by </b><i><a class="txt" href="circuit1.html#E9:31"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="circuit1.html#E12:31"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="circuit1.html#E13:31"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="circuit1.html#E16:31"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E41:31"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c17">y''</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c16">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E34:31"><i><font color="Green" title="E31">A27</font></i></a>, <a class="txt" href="circuit1.html#E38:31"><i><font color="Green" title="E33">A29</font></i></a>, <a class="txt" href="circuit1.html#E40:31"><i><font color="Green" title="E34">A30</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E42:31"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E43:31"/>
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="circuit1.html#E3:31"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="circuit1.html#E7:31"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="circuit1.html#E10:31"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="circuit1.html#E14:31"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="circuit1.html#E17:31"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>


</div>
