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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 (  for <font color="Olive" title="b3">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="b3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b3">C</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Olive" title="b1">t</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">C</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Olive" title="b1">t</font> holds <br/><font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> ) holds <br/> for <font color="Olive" title="b3">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>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b3">s</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Olive" title="b1">t</font> holds <br/> <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="Olive" title="b3">s</font>,<font color="Olive" title="b1">t</font>)</span>)</span> is    <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><br/><span class="kw">let </span><font color="Maroon" title="c2">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 (  for <font color="Olive" title="b2">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="b2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">C</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Olive" title="b1">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font> holds <br/><font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> ) holds <br/> for <font color="Olive" title="b2">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>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font> holds <br/> <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="Olive" title="b2">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> is    <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><br/><span class="kw">let </span><font color="Maroon" title="c3">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"> ( (  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="c3">A</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="c2">t</font> &amp; <font color="Maroon" title="c3">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">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="c2">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="c3">A</font> ) implies  for <font color="Olive" title="b1">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>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">s</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font> holds <br/> <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="Olive" title="b1">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> is    <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><br/>





<span class="kw">assume </span><a NAME="E1:116"/><span class="lab"><font color="Green" title="E66">A1</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="c3">A</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="c2">t</font> &amp; <font color="Maroon" title="c3">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">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="c2">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="c3">A</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> 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>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">s</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font> holds <br/> <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="Olive" title="b1">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> is    <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><br/>

<span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <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>; <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"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">s</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font> implies  <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="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> is    <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><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:116"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E3:116"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c2">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">  <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="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> is    <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><br/>

<a NAME="E4:116"/><span class="lab"><font color="Green" title="E69">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></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> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</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="kw">hence </span><a NAME="E5:116"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D2" onmouseover="rs('rewrite1/D2')" onmouseout="rh()">REWRITE1:def 2</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">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> or  not <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> or  not <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:116"/><span class="lab"><font color="Green" title="E70">A5</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:116"/><span class="lab"><font color="Green" title="E71">A6</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a> </span><br/>

<a NAME="E8:116"/><span class="lab"><font color="Green" title="E72">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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="c4">s</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#D19" target="_self" onmouseover="rs('abcmiz_0/D19')" onmouseout="rh()">Def19</a></span>;<br/>
<a NAME="E9:116"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A8</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:116"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="ref" href="finseq_5.html#D3" onmouseover="rs('finseq_5/D3')" onmouseout="rh()">FINSEQ_5:def 3</a></span>;<br/>
<a NAME="E10:116"/>
<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E69">A4</font></span></a>, <a class="txt" href="#E8:116"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="txt" href="#E7:116"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:116"/><span class="lab"><font color="Green" title="E74">A9</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/>
<a NAME="E13:116"/><span class="lab"><font color="Green" title="E75">A10</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:116"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="txt" href="#E7:116"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="ref" href="finseq_5.html#D3" onmouseover="rs('finseq_5/D3')" onmouseout="rh()">FINSEQ_5:def 3</a></span>;<br/>
<a NAME="E14:116"/><span class="lab"><font color="Green" title="E76">A11</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:116"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E15:116"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:116"><span class="lab"><font color="Green" title="E74">A9</font></span></a></span>;<br/>
<a NAME="E16:116"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A12</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:116"><span class="lab"><font color="Green" title="E76">A11</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E17:116"/>
<font color="Maroon" title="c6">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:116"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="#E1:116"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E3:116"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="txt" href="#E9:116"><span class="lab"><font color="Green" title="E73">A8</font></span></a>, <a class="txt" href="#E13:116"><span class="lab"><font color="Green" title="E75">A10</font></span></a>, <a class="txt" href="#E12:116"><span class="lab"><font color="Green" title="E74">A9</font></span></a>, <a class="txt" href="#E16:116"><span class="lab"><font color="Green" title="E77">A12</font></span></a>, <a class="ref" href="abcmiz_0.html#T70" target="_self" onmouseover="rs('abcmiz_0/T70')" onmouseout="rh()">Th70</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
