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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W1</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">W2</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font><br/> for <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> holds <br/><font color="Olive" title="b1">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Olive" title="b3">m</font>,<font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Olive" title="b3">m</font>,<font color="Olive" title="b4">n</font></span><br/><b>let </b><font color="Maroon" title="c3">W1</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W2</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font><br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W2</font> holds <br/><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">n</font></span><br/><b>let </b><font color="Maroon" title="c4">W2</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> holds <br/><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font></span><br/><b>let </b><font color="Maroon" title="c5">m</font>, <font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> implies <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> )</span><br/>









<b>assume </b><a NAME="E1:172"/><i><font color="Green" title="E119">A1</font></i>: 
<font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="172_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:172_1_1"/>
( ( not <font color="Maroon" title="c5">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; not <font color="Maroon" title="c6">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> ) or <font color="Maroon" title="c5">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or <font color="Maroon" title="c6">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> or  not <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W1</font> or  not <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="172_1_1_1"><b>suppose </b></a><a NAME="E1:172_1_1_1"/><i><font color="Green" title="E120">A2</font></i>: 
( not <font color="Maroon" title="c5">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; not <font color="Maroon" title="c6">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font></span><br/><div class="add"><a NAME="E2:172_1_1_1"/><b>then </b><i><font color="Green" title="E121">A3</font></i>: 
<font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c5">m</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c6">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self" title="GLIB_001:def.12">Def12</a></i>;<br/><a NAME="E3:172_1_1_1"/>
( <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c6">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c6">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:172"><i><font color="Green" title="E119">A1</font></i></a>, <a class="ref" href="glib_001.html#T49" target="_self" title="GLIB_001:th.49">Th49</a></i>;<br/><a NAME="E4:172_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c5">m</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c6">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:172_1_1_1"><i><font color="Green" title="E121">A3</font></i></a>, <a class="ref" href="glib_001.html#T36" target="_self" title="GLIB_001:th.36">Th36</a></i>;<br/><b>hence </b><a NAME="E5:172_1_1_1"/>
<font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="#E1:172"><i><font color="Green" title="E119">A1</font></i></a>, <a class="txt" href="#E1:172_1_1_1"><i><font color="Green" title="E120">A2</font></i></a>, <a class="ref" href="glib_001.html#D12" target="_self" title="GLIB_001:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="172_1_1_2"><b>suppose </b></a><a NAME="E1:172_1_1_2"/><i><font color="Green" title="E120">A4</font></i>: 
( <font color="Maroon" title="c5">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or <font color="Maroon" title="c6">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> or  not <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W1</font> or  not <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> = 
<font color="Maroon" title="c4">W2</font>
<b>by </b><i><a class="txt" href="#E1:172"><i><font color="Green" title="E119">A1</font></i></a>, <a class="ref" href="glib_001.html#D12" target="_self" title="GLIB_001:def.12">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font>
<b>by </b><i><a class="txt" href="#E1:172"><i><font color="Green" title="E119">A1</font></i></a>, <a class="txt" href="#E1:172_1_1_2"><i><font color="Green" title="E120">A4</font></i></a>, <a class="ref" href="glib_001.html#D12" target="_self" title="GLIB_001:def.12">Def12</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:172"/>
<font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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