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<span class="kw">let </span><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="struct_0.html#V11" title="STRUCT_0:attr.11">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</font><br/>  for <font color="Olive" title="b2">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font> holds <br/> (  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff ( <font color="Olive" title="b2">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="Olive" title="b2">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> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font> holds <br/> (  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> (<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff ( <font color="Olive" title="b1">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="Olive" title="b1">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> ) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff ( <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> ) )</span><br/>





<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:30"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<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#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">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> 
 <span class="kw">and </span><br/><a NAME="E3:30"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <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="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c4">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#D6" target="_self" title="CIRCUIT1:def.6">Def6</a></span>;<br/>
<span class="kw">reconsider </span><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="E5:30"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2: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#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E7:30"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c6">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#K5" title="CARD_1:func.5">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#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">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> 
 <span class="kw">and </span><br/><a NAME="E8:30"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 <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="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">ss</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#D4" target="_self" title="CIRCUIT1:def.4">Def4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">YY</font> = <font color="Maroon" title="c6">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/>
<span class="kw">consider </span><font color="Maroon" title="c8">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">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><span class="kw"> such that </span><br/><a NAME="E11:30"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">t99</font> = <font color="Maroon" title="c8">t</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">t2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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#U3" title="MSUALG_1:sel.3">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><span class="kw"> such that </span><br/><a NAME="E14:30"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t2</font>
 <span class="kw">and </span><br/><a NAME="E15:30"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K8" title="MSAFREE2:func.8">depth</a> <font color="Maroon" title="c10">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#D5" target="_self" title="CIRCUIT1:def.5">Def5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">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="c12">t9</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><span class="kw"> such that </span><br/><a NAME="E17:30"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
( <font color="Maroon" title="c11">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t2</font> &amp; <font color="Maroon" title="c12">t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">dt</font> )
 <span class="kw">and </span><br/><a NAME="E18:30"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
 <a href="msafree2.html#K8" title="MSAFREE2:func.8">depth</a> <font color="Maroon" title="c10">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="c12">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">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> <span class="kw"> such that </span><br/><a NAME="E20:30"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">t9</font>
 <span class="kw">and </span><br/><a NAME="E21:30"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">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="c12">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">y99</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E23:30"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">y99</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="c9">t99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E17:30"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E20:30"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E24:30"/>
(  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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <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> ) implies  <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
 <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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>

<a NAME="E2:30_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
 for <font color="Olive" title="b1">kk</font> being   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-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="c7">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="30_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">kk</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">YY</font> implies <font color="Maroon" title="c15">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c15">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">YY</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">tt</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">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><span class="kw"> such that </span><br/><a NAME="E3:30_1_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">kk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">tiv</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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#U3" title="MSUALG_1:sel.3">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><span class="kw"> such that </span><br/><a NAME="E5:30_1_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
( <font color="Maroon" title="c16">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">tiv</font> &amp;  <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c16">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K8" title="MSAFREE2:func.8">depth</a> <font color="Maroon" title="c17">tiv</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#D5" target="_self" title="CIRCUIT1:def.5">Def5</a></span>;<br/>
<a NAME="E6:30_1_1"/>
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c16">tt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>;<br/>
<a NAME="E7:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A18</font></span>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c16">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>;<br/>
<a NAME="E8:30_1_1"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
 ex <font color="Olive" title="b1">dt9</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> ex <font color="Olive" title="b2">t999</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> st <br/>( <font color="Olive" title="b1">dt9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">tiv</font> &amp; <font color="Olive" title="b2">t999</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">dt9</font> &amp;  <a href="msafree2.html#K8" title="MSAFREE2:func.8">depth</a> <font color="Maroon" title="c17">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="Olive" title="b2">t999</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></span>;<br/>
<a NAME="E9:30_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">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="c16">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E5:30_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="funct_1.html#T4" title="FUNCT_1:th.4">FUNCT_1:4</a>, <a class="ref" href="trees_1.html#T29" title="TREES_1:th.29">TREES_1:29</a>, <a class="ref" href="trees_1.html#T43" title="TREES_1:th.43">TREES_1:43</a></span>;<br/>
<a NAME="E10:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">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="c16">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E5:30_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E8:30_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="pre_circ.html#T1" title="PRE_CIRC:th.1">PRE_CIRC:1</a>, <a class="ref" href="trees_1.html#T29" title="TREES_1:th.29">TREES_1:29</a>, <a class="ref" href="trees_1.html#T43" title="TREES_1:th.43">TREES_1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:30_1_1"/>
<font color="Maroon" title="c15">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="card_1.html#T30" title="CARD_1:th.30">CARD_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:30_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">t99</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="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E14:30"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E15:30"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E17:30"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E18:30"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="funct_1.html#T4" title="FUNCT_1:th.4">FUNCT_1:4</a>, <a class="ref" href="trees_1.html#T29" title="TREES_1:th.29">TREES_1:29</a>, <a class="ref" href="trees_1.html#T43" title="TREES_1:th.43">TREES_1:43</a></span>;<br/>
<a NAME="E4:30_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t99</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="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E14:30"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E15:30"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E17:30"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E18:30"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="pre_circ.html#T1" title="PRE_CIRC:th.1">PRE_CIRC:1</a>, <a class="ref" href="trees_1.html#T29" title="TREES_1:th.29">TREES_1:29</a>, <a class="ref" href="trees_1.html#T43" title="TREES_1:th.43">TREES_1:43</a></span>;<br/>
<a NAME="E5:30_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" title="CARD_1:th.30">CARD_1:30</a></span>;<br/>
<a NAME="E6:30_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">YY</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<a NAME="E7:30_1"/><span class="kw">then </span>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>;<br/>
<a NAME="E8:30_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T10" target="_self" title="CIRCUIT1:th.10">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:30_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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E25:30"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">ct</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">t</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E27:30"/>
 <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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T22" title="TREES_1:th.22">TREES_1:22</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E29:30"/><span class="lab"><font color="Green" title="E29">A20</font></span>: 
<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="c9">t99</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<a NAME="E30:30"/><span class="lab"><font color="Green" title="E30">A21</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">ss</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E31:30"/>
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E32:30"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E33:30"/><span class="kw">then </span>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T10" target="_self" title="CIRCUIT1:th.10">Th11</a></span>;<br/>
<a NAME="E34:30"/><span class="kw">then </span>
<font color="Maroon" title="c15">ct</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E30:30"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>;<br/>
<a NAME="E35:30"/><span class="kw">then </span>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E37:30"/><span class="lab"><font color="Green" title="E31">A22</font></span>: 
<font color="Maroon" title="c8">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="c17">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T42" title="CARD_2:th.42">CARD_2:42</a></span>;<br/>
<a NAME="E38:30"/>
<font color="Maroon" title="c17">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:30"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="txt" href="#E29:30"><span class="lab"><font color="Green" title="E29">A20</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E39:30"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">y99</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:30"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="txt" href="#E23:30"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E40:30"/><span class="kw">then </span>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E41:30"/>
 <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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E15:30"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E18:30"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E21:30"><span class="lab"><font color="Green" title="E27">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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