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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>   <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a>   non  <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>   <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a>   <a href="abcmiz_0.html#L3" title="ABCMIZ_0:struct.3">TAS-structure</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">t</font> being   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">A</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">A</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Olive" title="b1">t</font> holds <br/><font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b2">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b1">t</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">R</font> = <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</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">A</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c3">t</font> holds <br/><font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c3">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</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="c4">A</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c3">t</font> implies <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c3">t</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:117"/>
<font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c3">t</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="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c3">t</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">A9</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E3:117"/>
<font color="Maroon" title="c5">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E4:117"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
<font color="Maroon" title="c5">A9</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">and </span><br/><a NAME="E5:117"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A9</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">and </span><br/><a NAME="E6:117"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A9</font> &amp; <font color="Olive" title="b1">C</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font> holds <br/><font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#T64" target="_self" onmouseover="rs('abcmiz_0/T64')" onmouseout="rh()">Th64</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">s</font> being   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E8:117"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A9</font>
 <span class="kw">and </span><br/><a NAME="E9:117"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="abcmiz_0.html#T65" target="_self" onmouseover="rs('abcmiz_0/T65')" onmouseout="rh()">Th65</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> =  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> as    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E8:117"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="txt" href="#E9:117"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="abcmiz_0.html#T71" target="_self" onmouseover="rs('abcmiz_0/T71')" onmouseout="rh()">Th71</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">p</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D3" onmouseover="rs('rewrite1/D3')" onmouseout="rh()">REWRITE1:def 3</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="c7">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="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c7">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="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <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> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">t</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T62" onmouseover="rs('finseq_5/T62')" onmouseout="rh()">FINSEQ_5:62</a></span>
<br/>.= 
<font color="Maroon" title="c6">s</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c3">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#D19" target="_self" onmouseover="rs('abcmiz_0/D19')" onmouseout="rh()">Def19</a></span>
<br/>.= 
<font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E8:117"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="txt" href="#E9:117"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="abcmiz_0.html#T56" target="_self" onmouseover="rs('abcmiz_0/T56')" onmouseout="rh()">Th56</a>, <a class="ref" href="abcmiz_0.html#T57" target="_self" onmouseover="rs('abcmiz_0/T57')" onmouseout="rh()">Th57</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"> <font color="Maroon" title="c7">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="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c7">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="c7">p</font></span>)</span> = 
<font color="Maroon" title="c7">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> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">t</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" onmouseover="rs('finseq_5/D3')" onmouseout="rh()">FINSEQ_5:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T62" onmouseover="rs('finseq_5/T62')" onmouseout="rh()">FINSEQ_5:62</a></span>
<br/>.= 
<font color="Maroon" title="c3">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#D19" target="_self" onmouseover="rs('abcmiz_0/D19')" onmouseout="rh()">Def19</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/>


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