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<b>let </b><font color="Maroon" title="c1">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">S</font> being  <a href="rewrite2.html#NM1" title="REWRITE2:NM.1">semi-Thue-system</a> of <font color="Maroon" title="c1">E</font><br/> for <font color="Olive" title="b2">u</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> <br/> for <font color="Olive" title="b3">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>   st <font color="Olive" title="b3">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Olive" title="b1">S</font> holds <br/>( <font color="Olive" title="b3">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Olive" title="b2">u</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b2">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Olive" title="b3">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Olive" title="b1">S</font> )</span><br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="rewrite2.html#NM1" title="REWRITE2:NM.1">semi-Thue-system</a> of <font color="Maroon" title="c1">E</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">u</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> <br/> for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>   st <font color="Olive" title="b2">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Olive" title="b1">u</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Olive" title="b2">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/><b>let </b><font color="Maroon" title="c3">u</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</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">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>   st <font color="Olive" title="b1">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Olive" title="b1">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/><b>let </b><font color="Maroon" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> implies ( <font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> ) )</span><br/>







<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c4">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/>

<a NAME="E2:43"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> )
 
<b>by </b><i><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></i>;<br/>
<a NAME="E3:43"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</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="c5">i</font> <a href="hidden.html#R2" title="HIDDEN: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">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN: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">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/><b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN: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">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN: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">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span><br/><a NAME="E2:43_1"/><i><font color="Green" title="E28">A4</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="#E1:43_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></i>;<br/><a NAME="E3:43_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/><a NAME="E4:43_1"/><b>then </b><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></i>;<br/><a NAME="E5:43_1"/><i><font color="Green" title="E30">A6</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">u</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">u</font> )
 <b>by </b><i><a class="txt" href="#E2:43_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></i>;<br/><a NAME="E6:43_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E4:43_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="rewrite2.html#T12" target="_self" title="REWRITE2:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E7:43_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E5:43_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</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>
<b>hence </b><a NAME="E5:43"/>
<font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K7" title="REWRITE2:func.7">+^</a> <font color="Maroon" title="c3">u</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E3:43"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</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="c5">i</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN: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="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/><b>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E27">A7</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN: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="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span><br/><a NAME="E2:43_2"/><i><font color="Green" title="E28">A8</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="#E1:43_2"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></i>;<br/><a NAME="E3:43_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/><a NAME="E4:43_2"/><b>then </b><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></i>;<br/><a NAME="E5:43_2"/><i><font color="Green" title="E30">A10</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:43_2"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></i>;<br/><a NAME="E6:43_2"/>
<font color="Maroon" title="c3">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c3">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E4:43_2"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="rewrite2.html#T12" target="_self" title="REWRITE2:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E7:43_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rewrite2.html#K4" title="REWRITE2:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E5:43_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</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>
<b>hence </b><a NAME="E7:43"/>
<font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">^+</a> <font color="Maroon" title="c4">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E3:43"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</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>
