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<span class="kw">let </span><font color="Maroon" title="c1">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">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">s</font>, <font color="Olive" title="b3">t</font>, <font color="Olive" title="b4">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>   st <font color="Olive" title="b2">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b3">t</font>,<font color="Olive" title="b1">S</font> holds <br/>( <font color="Olive" title="b2">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b4">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b3">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b4">u</font>,<font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b4">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b4">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">t</font>,<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" 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">s</font>, <font color="Olive" title="b2">t</font>, <font color="Olive" title="b3">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>   st <font color="Olive" title="b1">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b2">t</font>,<font color="Maroon" title="c2">S</font> holds <br/>( <font color="Olive" title="b1">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b2">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">u</font>,<font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b3">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b1">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b3">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">t</font>,<font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font>, <font color="Maroon" title="c5">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</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="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font> implies ( <font color="Maroon" title="c3">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c4">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:51"/>
<font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</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="c3">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c4">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font> )</span><br/>

<a NAME="E2:51"/><span class="kw">then </span>
 <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E4:51"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E5:51"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D3" onmouseover="rs('rewrite1/D3')" onmouseout="rh()">REWRITE1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c6">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="abcmiz_0.html#T73" onmouseover="rs('abcmiz_0/T73')" onmouseout="rh()">ABCMIZ_0:73</a></span>;<br/>
<a NAME="E7:51"/>
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="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" onmouseover="rs('finseq_5/T6')" onmouseout="rh()">FINSEQ_5:6</a></span>;<br/>
<a NAME="E8:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">+^</a> <font color="Maroon" title="c5">u</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="rewrite2.html#D3" target="_self" onmouseover="rs('rewrite2/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E9:51"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">+^</a> <font color="Maroon" title="c5">u</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" onmouseover="rs('rewrite2/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E10:51"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">+^</a> <font color="Maroon" title="c5">u</font></span>)</span> <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="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" onmouseover="rs('finseq_5/T6')" onmouseout="rh()">FINSEQ_5:6</a></span>;<br/>
<a NAME="E11:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">+^</a> <font color="Maroon" title="c5">u</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">.</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="c7">p</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">+^</a> <font color="Maroon" title="c5">u</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E9:51"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="rewrite2.html#D3" target="_self" onmouseover="rs('rewrite2/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E12:51"/>
<font color="Maroon" title="c7">p</font> <a href="rewrite2.html#K5" title="REWRITE2:func.5">+^</a> <font color="Maroon" title="c5">u</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T23" target="_self" onmouseover="rs('rewrite2/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E13:51"/><span class="kw">then </span>
 <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c4">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="rewrite1.html#D3" onmouseover="rs('rewrite1/D3')" onmouseout="rh()">REWRITE1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:51"/>
<font color="Maroon" title="c3">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c4">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">S</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">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E15:51"/>
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="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" onmouseover="rs('finseq_5/T6')" onmouseout="rh()">FINSEQ_5:6</a></span>;<br/>
<a NAME="E16:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">^+</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="rewrite2.html#D2" target="_self" onmouseover="rs('rewrite2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E17:51"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">^+</a> <font color="Maroon" title="c7">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" onmouseover="rs('rewrite2/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E18:51"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">^+</a> <font color="Maroon" title="c7">p</font></span>)</span> <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="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" onmouseover="rs('finseq_5/T6')" onmouseout="rh()">FINSEQ_5:6</a></span>;<br/>
<a NAME="E19:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">^+</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">.</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="c5">u</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">^+</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c4">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E17:51"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="rewrite2.html#D2" target="_self" onmouseover="rs('rewrite2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E20:51"/>
<font color="Maroon" title="c5">u</font> <a href="rewrite2.html#K4" title="REWRITE2:func.4">^+</a> <font color="Maroon" title="c7">p</font> is    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T23" target="_self" onmouseover="rs('rewrite2/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E21:51"/><span class="kw">then </span>
 <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c4">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E19:51"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="rewrite1.html#D3" onmouseover="rs('rewrite1/D3')" onmouseout="rh()">REWRITE1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:51"/>
<font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</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"> verum</span><br/>


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