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<span class="kw">let </span><font color="Maroon" title="c1">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">E</font><br/>  for <font color="Olive" title="b3">W2</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font><br/>  for <font color="Olive" title="b4">W1</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b2">G1</font>  st <font color="Olive" title="b1">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b4">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W2</font> &amp; <font color="Olive" title="b3">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">E</font> holds <br/>( <font color="Olive" title="b4">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  iff <font color="Olive" title="b3">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</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="c1">G2</font><br/>  for <font color="Olive" title="b3">W1</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b1">G1</font>  st <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> &amp; <font color="Olive" title="b2">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">E</font> holds <br/>( <font color="Olive" title="b3">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  iff <font color="Olive" title="b2">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">G1</font> be    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c1">G2</font><br/>  for <font color="Olive" title="b2">W1</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G1</font>  st <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b2">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W2</font> &amp; <font color="Olive" title="b1">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">E</font> holds <br/>( <font color="Olive" title="b2">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  iff <font color="Olive" title="b1">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  )</span><br/><span class="kw">let </span><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="c1">G2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">G1</font>  st <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">E</font> holds <br/>( <font color="Olive" title="b1">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  iff <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">W1</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c5">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">E</font> implies ( <font color="Maroon" title="c5">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  iff <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">and </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c5">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font>
 <span class="kw">and </span><br/><a NAME="E3:20"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">E</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  iff <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  )</span><br/>

<a NAME="E4:20"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W2</font> holds <br/><font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="glib_001.html#T100" onmouseover="rs('glib_001/T100')" onmouseout="rh()">GLIB_001:100</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  implies <font color="Maroon" title="c5">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:20_1"/>
<font color="Maroon" title="c5">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> </span><br/><a NAME="E2:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A5</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W1</font> holds <br/><font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T122" onmouseover="rs('glib_001/T122')" onmouseout="rh()">GLIB_001:122</a></span>;<br/><a NAME="E3:20_1"/>
 for <font color="Olive" title="b1">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1_1"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G2</font></span><br/>

<a NAME="E2:20_1_1"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="glib_001.html#T1" onmouseover="rs('glib_001/T1')" onmouseout="rh()">GLIB_001:1</a></span>;<br/>
<a NAME="E3:20_1_1"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/>
<a NAME="E4:20_1_1"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E5:20_1_1"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E6:20_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E7:20_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T21" onmouseover="rs('glib_001/T21')" onmouseout="rh()">GLIB_001:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">m</font> = <span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">m</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E10:20_1_1"/>
( <font color="Maroon" title="c6">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E3:20_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a>, <a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E11:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">a12</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">W2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T25" onmouseover="rs('glib_001/T25')" onmouseout="rh()">GLIB_001:25</a></span>;<br/>
<a NAME="E12:20_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c5">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E21">A2</font></span></a></span>;<br/>
<a NAME="E13:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T1" onmouseover="rs('glib_001/T1')" onmouseout="rh()">GLIB_001:1</a></span>;<br/>
<a NAME="E14:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/>
<a NAME="E15:20_1_1"/>
<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c3">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E13:20_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:20_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E11:20_1_1"><span class="lab"><font color="Green" title="E30">a12</font></span></a>, <a class="txt" href="#E14:20_1_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="glib_007.html#T7" target="_self" onmouseover="rs('glib_007/T7')" onmouseout="rh()">Th7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><span class="kw">hence </span><a NAME="E4:20_1"/>
<font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T122" onmouseover="rs('glib_001/T122')" onmouseout="rh()">GLIB_001:122</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

<span class="kw">assume </span><a NAME="E6:20"/>
<font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> </span><br/>

<a NAME="E7:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A15</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T122" onmouseover="rs('glib_001/T122')" onmouseout="rh()">GLIB_001:122</a></span>;<br/>
<a NAME="E8:20"/>
 for <font color="Olive" title="b1">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W1</font> holds <br/><font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c3">G1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W1</font> implies <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c3">G1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_2"/><span class="lab"><font color="Green" title="E25">A16</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c3">G1</font></span><br/>

<a NAME="E2:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A17</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E21">A2</font></span></a></span>;<br/>
<a NAME="E3:20_2"/><span class="lab"><font color="Green" title="E27">A18</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="glib_001.html#T1" onmouseover="rs('glib_001/T1')" onmouseout="rh()">GLIB_001:1</a></span>;<br/>
<a NAME="E4:20_2"/><span class="lab"><font color="Green" title="E28">A19</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/>
<a NAME="E5:20_2"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E6:20_2"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E25">A16</font></span></a></span>;<br/>
<a NAME="E7:20_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">m</font> = <span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">m</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E10:20_2"/>
( <font color="Maroon" title="c6">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">W2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="txt" href="#E4:20_2"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a>, <a class="txt" href="#E5:20_2"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E6:20_2"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E11:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">a23</font></span>: 
( <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> &amp; <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <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#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T24" onmouseover="rs('glib_001/T24')" onmouseout="rh()">GLIB_001:24</a></span>;<br/>
<a NAME="E12:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A23</font></span>: 
( <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> &amp; <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <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#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E21">A2</font></span></a></span>;<br/>
<a NAME="E13:20_2"/>
<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20_2"><span class="lab"><font color="Green" title="E31">a23</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E14:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A24</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T1" onmouseover="rs('glib_001/T1')" onmouseout="rh()">GLIB_001:1</a></span>;<br/>
<a NAME="E15:20_2"/><span class="lab"><font color="Green" title="E34">A25</font></span>: 
<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E26">A17</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/>
<a NAME="E16:20_2"/>
<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c1">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_2"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="txt" href="#E14:20_2"><span class="lab"><font color="Green" title="E33">A24</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:20_2"/>
<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c5">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E15:20_2"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="ref" href="glib_007.html#T7" target="_self" onmouseover="rs('glib_007/T7')" onmouseout="rh()">Th7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E9:20"/>
<font color="Maroon" title="c5">W1</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T122" onmouseover="rs('glib_001/T122')" onmouseout="rh()">GLIB_001:122</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
