<?xml version="1.0"?>
<div class="add">

<b>let </b><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> ;<br/><b>let </b><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>;<br/><b>let </b><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>;<br/>





<b>assume </b><a NAME="E1:120"/><i><font color="Green" title="E74">A1</font></i>: 
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>
 ;<br/>

<b>let </b><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>;<br/>

<b>assume </b><a NAME="E2:120"/><i><font color="Green" title="E75">A2</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> )
 ;<br/>

<a NAME="E3:120"/><i><font color="Green" title="E76">A3</font></i>: 
 <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#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">j</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E4:120"/><i><font color="Green" title="E77">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</font> <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> )
 ;<br/>

<a NAME="E5:120"/><b>then </b>
( <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">j</font> is    <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>  )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:120"/><b>then </b><i><font color="Green" title="E78">A5</font></i>: 
( <font color="Maroon" title="c5">j</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">s</font> &amp; <font color="Maroon" title="c5">j</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"><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> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:120"><i><font color="Green" title="E76">A3</font></i></a>, <a class="txt" href="abcmiz_0.html#E4:120"><i><font color="Green" title="E77">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:120"/><b>then </b>
( <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font> as   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E9:120"/>
( <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> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E6:120"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">tt</font> = <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> <font color="Maroon" title="c5">j</font> as   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E11:120"/><i><font color="Green" title="E79">A6</font></i>: 
( <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"><font color="Maroon" title="c5">j</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> <font color="Maroon" title="c6">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c7">tt</font> &amp; <font color="Maroon" title="c6">a</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Maroon" title="c7">tt</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:120"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:120"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a>, <a class="ref" href="abcmiz_0.html#D28" target="_self" title="ABCMIZ_0:def.28">Def28</a></i>;<br/>
<a NAME="E12:120"/>
not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c7">tt</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:120_1"/><i><font color="Green" title="E80">A7</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c7">tt</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:120_1"/><i><font color="Green" title="E81">A8</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:120"><i><font color="Green" title="E77">A4</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">i</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">s1</font> = <font color="Maroon" title="c4">s</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">i</font></span>)</span> as    <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> <b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E6:120_1"/>
<font color="Maroon" title="c10">s1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#D1" title="TREES_1:def.1">TREES_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">s2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E8:120_1"/><i><font color="Green" title="E82">A9</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c11">s2</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">s2</font> = <font color="Maroon" title="c11">s2</font> as    <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> <b>by </b><i><a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">s21</font> = <font color="Maroon" title="c12">s2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span> as    <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> <b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E11:120_1"/>
<font color="Maroon" title="c9">i</font> <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>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E4:120"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:120_1"><i><font color="Green" title="E81">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E12:120_1"/><b>then </b><i><font color="Green" title="E83">A10</font></i>: 
(  <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">s2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E13:120_1"/><b>then </b><i><font color="Green" title="E84">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">s2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E4:120"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:120_1"><i><font color="Green" title="E81">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:120_1"/><b>then </b><i><font color="Green" title="E85">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">s21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E15:120_1"/><b>then </b><i><font color="Green" title="E86">A13</font></i>: 
<font color="Maroon" title="c13">s21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">s21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E16:120_1"/><b>then </b><i><font color="Green" title="E87">A14</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">s21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">s21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c13">s21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E18:120_1"/>
<font color="Maroon" title="c13">s21</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c12">s2</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#D1" title="TREES_1:def.1">TREES_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">s22</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E20:120_1"/><i><font color="Green" title="E88">A15</font></i>: 
<font color="Maroon" title="c12">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s21</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c15">s22</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">s22</font> = <font color="Maroon" title="c15">s22</font> as    <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> <b>by </b><i><a class="txt" href="abcmiz_0.html#E20:120_1"><i><font color="Green" title="E88">A15</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E22:120_1"/><i><font color="Green" title="E89">A16</font></i>: 
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="c12">s2</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E13:120_1"><i><font color="Green" title="E84">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green" title="E90">A17</font></i>: <font color="Maroon" title="c14">b</font> = 
<font color="Maroon" title="c12">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="abcmiz_0.html#E15:120_1"><i><font color="Green" title="E86">A13</font></i></a>, <a class="txt" href="abcmiz_0.html#E20:120_1"><i><font color="Green" title="E88">A15</font></i></a>, <a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></i>
<br/>.= 
<font color="Maroon" title="c6">a</font>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:120_1"><i><font color="Green" title="E81">A8</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="txt" href="abcmiz_0.html#E12:120_1"><i><font color="Green" title="E83">A10</font></i></a>, <a class="txt" href="abcmiz_0.html#E22:120_1"><i><font color="Green" title="E89">A16</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/>
<a NAME="E24:120_1"/><i><font color="Green" title="E91">A18</font></i>: 
<font color="Maroon" title="c10">s1</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:120"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="txt" href="abcmiz_0.html#E12:120_1"><i><font color="Green" title="E83">A10</font></i></a>, <a class="ref" href="abcmiz_0.html#T61" target="_self" title="ABCMIZ_0:th.61">Th61</a></i>;<br/>
<a NAME="E25:120_1"/>
<font color="Maroon" title="c10">s1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">tt</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:120_1"><i><font color="Green" title="E81">A8</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="txt" href="abcmiz_0.html#E12:120_1"><i><font color="Green" title="E83">A10</font></i></a>, <a class="ref" href="abcmiz_0.html#T37" target="_self" title="ABCMIZ_0:th.37">Th37</a></i>;<br/>
<b>then </b><i><font color="Green" title="E92">A19</font></i>: <font color="Maroon" title="c10">s1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> = 
<font color="Maroon" title="c6">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E1:120_1"><i><font color="Green" title="E80">A7</font></i></a>, <a class="ref" href="abcmiz_0.html#T25" target="_self" title="ABCMIZ_0:th.25">Th25</a></i>
<br/>.= 
<font color="Maroon" title="c13">s21</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E15:120_1"><i><font color="Green" title="E86">A13</font></i></a>, <a class="txt" href="abcmiz_0.html#E23:120_1"><i><font color="Green" title="E90">A17</font></i></a>, <a class="ref" href="abcmiz_0.html#T32" target="_self" title="ABCMIZ_0:th.32">Th32</a></i>
;<br/>
<a NAME="E27:120_1"/>
<font color="Maroon" title="c12">s2</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c10">s1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:120"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="ref" href="abcmiz_0.html#T61" target="_self" title="ABCMIZ_0:th.61">Th61</a></i>;<br/>
<a NAME="E28:120_1"/><b>then </b>
<font color="Maroon" title="c16">s22</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c10">s1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E20:120_1"><i><font color="Green" title="E88">A15</font></i></a>, <a class="txt" href="abcmiz_0.html#E26:120_1"><i><font color="Green" title="E92">A19</font></i></a>, <a class="ref" href="abcmiz_0.html#T61" target="_self" title="ABCMIZ_0:th.61">Th61</a></i>;<br/>
<a NAME="E29:120_1"/><b>then </b><i><font color="Green" title="E93">A20</font></i>: 
<font color="Maroon" title="c10">s1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">s22</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E24:120_1"><i><font color="Green" title="E91">A18</font></i></a>, <a class="ref" href="abcmiz_0.html#T62" target="_self" title="ABCMIZ_0:th.62">Th62</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">B</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">s22</font></span>)</span> as   <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> ;<br/>
<font color="Maroon" title="c4">s</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> = 
<font color="Maroon" title="c12">s2</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="ref" href="abcmiz_0.html#T38" target="_self" title="ABCMIZ_0:th.38">Th38</a></i>
<br/>.= 
<font color="Maroon" title="c16">s22</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E20:120_1"><i><font color="Green" title="E88">A15</font></i></a>, <a class="txt" href="abcmiz_0.html#E26:120_1"><i><font color="Green" title="E92">A19</font></i></a>, <a class="ref" href="abcmiz_0.html#T38" target="_self" title="ABCMIZ_0:th.38">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">s22</font></span>)</span> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font>
<b>by </b><i><a class="ref" href="abcmiz_0.html#T38" target="_self" title="ABCMIZ_0:th.38">Th38</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E94">A21</font></i>: <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> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c16">s22</font></span>)</span> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:120"><i><font color="Green" title="E75">A2</font></i></a>, <a class="ref" href="abcmiz_0.html#T57" target="_self" title="ABCMIZ_0:th.57">Th57</a>, <a class="ref" href="abcmiz_0.html#T58" target="_self" title="ABCMIZ_0:th.58">Th58</a></i>
<br/>.= 
<font color="Maroon" title="c17">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E29:120_1"><i><font color="Green" title="E93">A20</font></i></a>, <a class="ref" href="abcmiz_0.html#T57" target="_self" title="ABCMIZ_0:th.57">Th57</a>, <a class="ref" href="abcmiz_0.html#T58" target="_self" title="ABCMIZ_0:th.58">Th58</a></i>
;<br/>
<a NAME="E33:120_1"/><i><font color="Green" title="E95">A22</font></i>: 
<font color="Maroon" title="c17">B</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E29:120_1"><i><font color="Green" title="E93">A20</font></i></a>, <a class="ref" href="abcmiz_0.html#D29" target="_self" title="ABCMIZ_0:def.29">Def29</a></i>;<br/>
<a NAME="E34:120_1"/><i><font color="Green" title="E96">A23</font></i>: 
( <font color="Maroon" title="c17">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">s1</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">s22</font></span>)</span> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">s1</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">s2</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">s21</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">s22</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:120"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="txt" href="abcmiz_0.html#E20:120_1"><i><font color="Green" title="E88">A15</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E35:120_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">s22</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">s2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E36:120_1"/><b>then </b><i><font color="Green" title="E97">A24</font></i>: 
<font color="Maroon" title="c17">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E1:120"><i><font color="Green" title="E74">A1</font></i></a>, <a class="txt" href="abcmiz_0.html#E32:120_1"><i><font color="Green" title="E94">A21</font></i></a>, <a class="txt" href="abcmiz_0.html#E33:120_1"><i><font color="Green" title="E95">A22</font></i></a>, <a class="txt" href="abcmiz_0.html#E34:120_1"><i><font color="Green" title="E96">A23</font></i></a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E37:120_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">s21</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E16:120_1"><i><font color="Green" title="E87">A14</font></i></a>, <a class="txt" href="abcmiz_0.html#E23:120_1"><i><font color="Green" title="E90">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E38:120_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">s2</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E34:120_1"><i><font color="Green" title="E96">A23</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E39:120_1"/><b>then </b><i><font color="Green" title="E98">A25</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E34:120_1"><i><font color="Green" title="E96">A23</font></i></a>, <a class="txt" href="abcmiz_0.html#E36:120_1"><i><font color="Green" title="E97">A24</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_5"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">s1</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">s22</font> )
 </span><b>by </b><i><a class="txt" href="abcmiz_0.html#E34:120_1"><i><font color="Green" title="E96">A23</font></i></a>, <a class="txt" href="abcmiz_0.html#E39:120_1"><i><font color="Green" title="E98">A25</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_5_1"><b>suppose </b></a><a NAME="E1:120_1_5_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">s1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:120_1_5_1"/><i><font color="Green" title="E99">A26</font></i>: 
( <font color="Maroon" title="c18">x</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="c10">s1</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c18">x</font> as    <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>  <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:120_1_5_1"><i><font color="Green" title="E99">A26</font></i></a></i>;<br/><a NAME="E5:120_1_5_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c19">x</font> <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="c10">s1</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:120_1_5_1"><i><font color="Green" title="E99">A26</font></i></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:120_1_5_1"/><b>then </b>
( <font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c19">x</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">s</font> &amp; <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:120_1"><i><font color="Green" title="E81">A8</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="txt" href="abcmiz_0.html#E12:120_1"><i><font color="Green" title="E83">A10</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:120_1_5_1"><i><font color="Green" title="E99">A26</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E7:120_1_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:120"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_5_2"><b>suppose </b></a><a NAME="E1:120_1_5_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">s22</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:120_1_5_2"/><i><font color="Green" title="E99">A27</font></i>: 
( <font color="Maroon" title="c18">x</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="c16">s22</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s22</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c18">x</font> as    <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>  <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:120_1_5_2"><i><font color="Green" title="E99">A27</font></i></a></i>;<br/><a NAME="E5:120_1_5_2"/>
<font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:120_1_5_2"><i><font color="Green" title="E99">A27</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:120_1_5_2"/><b>then </b><i><font color="Green" title="E100">A28</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">x</font> &amp; <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:120_1"><i><font color="Green" title="E81">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:120_1_5_2"/>
( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">x</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="c12">s2</font> &amp; <font color="Maroon" title="c12">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E14:120_1"><i><font color="Green" title="E85">A12</font></i></a>, <a class="txt" href="abcmiz_0.html#E20:120_1"><i><font color="Green" title="E88">A15</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:120_1_5_2"><i><font color="Green" title="E99">A27</font></i></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E8:120_1_5_2"/><b>then </b>
( <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">x</font></span>)</span> <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">s</font> &amp; <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E8:120_1"><i><font color="Green" title="E82">A9</font></i></a>, <a class="txt" href="abcmiz_0.html#E12:120_1"><i><font color="Green" title="E83">A10</font></i></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E9:120_1_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:120"><i><font color="Green" title="E78">A5</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:120_1_5_2"><i><font color="Green" title="E100">A28</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E13:120"/>
<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="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="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon" title="c5">j</font></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> 
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E11:120"><i><font color="Green" title="E79">A6</font></i></a>, <a class="ref" href="abcmiz_0.html#D31" target="_self" title="ABCMIZ_0:def.31">Def31</a></i>;<br/>


</div>
