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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="CATALAN2:func.2">^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#K2" title="CATALAN2:func.2">^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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Olive" title="b1">S</font> holds <br/>( <font color="Olive" title="b3">p</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">+^</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b2">u</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Olive" title="b1">S</font> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="CATALAN2:func.2">^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#K2" title="CATALAN2:func.2">^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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">+^</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">u</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">let </span><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#K2" title="CATALAN2:func.2">^omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="CATALAN2:func.2">^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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">+^</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">let </span><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#K2" title="CATALAN2:func.2">^omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> implies ( <font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">+^</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c4">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">+^</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E26">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K6" title="REWRITE2:func.6">+^</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#K6" title="REWRITE2:func.6">+^</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#K6" title="REWRITE2:func.6">+^</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#K6" title="REWRITE2:func.6">+^</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<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#K6" title="REWRITE2:func.6">+^</a> <font color="Maroon" title="c3">u</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:42_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<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#K6" title="REWRITE2:func.6">+^</a> <font color="Maroon" title="c3">u</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K6" title="REWRITE2:func.6">+^</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#K6" title="REWRITE2:func.6">+^</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span><br/><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>;<br/><a NAME="E4:42_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">+^</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>;<br/><a NAME="E5:42_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>;<br/><a NAME="E6:42_1"/><span class="kw">then </span>
<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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>;<br/><a NAME="E7:42_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></span>;<br/><a NAME="E8:42_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T12" target="_self" title="REWRITE2:th.12">Th12</a></span>;<br/><a NAME="E9:42_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">+^</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_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#K6" title="REWRITE2:func.6">+^</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#K6" title="REWRITE2:func.6">+^</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E8:42_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E27">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="REWRITE2:func.5">^+</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#K5" title="REWRITE2:func.5">^+</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#K5" title="REWRITE2:func.5">^+</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#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<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#K5" title="REWRITE2:func.5">^+</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:42_2"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<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#K5" title="REWRITE2:func.5">^+</a> <font color="Maroon" title="c4">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K5" title="REWRITE2:func.5">^+</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#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span><br/><a NAME="E3:42_2"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_2"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>;<br/><a NAME="E4:42_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">^+</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>;<br/><a NAME="E5:42_2"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>;<br/><a NAME="E6:42_2"/><span class="kw">then </span>
<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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>;<br/><a NAME="E7:42_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></span>;<br/><a NAME="E8:42_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A15</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T12" target="_self" title="REWRITE2:th.12">Th12</a></span>;<br/><a NAME="E9:42_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">^+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_2"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_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#K5" title="REWRITE2:func.5">^+</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#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E8:42_2"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:42"/>
 <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#K6" title="REWRITE2:func.6">+^</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:42"/>
<font color="Maroon" title="c4">p</font> <a href="rewrite2.html#K6" title="REWRITE2:func.6">+^</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E6:42"/>
 <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#K5" title="REWRITE2:func.5">^+</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:42"/>
<font color="Maroon" title="c3">u</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">^+</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#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
