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<b>let </b><font color="Maroon">T</font> be  <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a> <a href="abcmiz_0.html#V1" target="_self">Noetherian</a> non <a href="abcmiz_0.html#V4" target="_self">void</a> <a href="abcmiz_0.html#V9" target="_self">adj-structured</a>  <a href="abcmiz_0.html#L3" target="_self">TAS-structure</a> ;<br/><b>let </b><font color="Maroon">t1</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>, <font color="Maroon">t2</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>;<br/>



<b>set </b><font color="Maroon">R</font> = <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> ;<br/>
<b>given </b><font color="Maroon">p</font> being    <a href="rewrite1.html#M1">RedSequence</a> of <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> <b> such that </b><a NAME="E1:126"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D3">REWRITE1:def 3</a><br/>

<a NAME="E2:126"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/>
<a NAME="E3:126"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:126"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">a</font> being  <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font> ex <font color="Olive">t</font> being  <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> &amp; <font color="Olive">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">a</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Olive">t</font> );<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="126_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:126_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font>
 ;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:126_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:126_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:126_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:126_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> 
 <b>by </b><i><a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>, <font color="Maroon">q2</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:126_1"/>
 ex <font color="Olive">a</font> being  <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font> st <br/>( not <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">q2</font> &amp; <font color="Olive">a</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">q2</font> &amp; <font color="Olive">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, </i>;<br/><b>hence </b><a NAME="E9:126_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:126"/><i><font color="Green">E50</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> )
 <b>and </b><br/><a NAME="E10:126"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<a NAME="E11:126"/>
<font color="Maroon">f</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> =  <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> implies <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E13:126"/><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E14:126"/><b>then </b>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D12" target="_self">Def12</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:126"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E16:126"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E110</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="126_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:126_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">j</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:126_2_1"/><i><font color="Green">E112</font></i>: 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:126_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:126_2_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:126_2_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E7:126_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E8:126_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">a</font> being   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font>, <font color="Maroon">t</font> being   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E10:126_2_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">t</font> )
 <b>by </b><i/>;<br/><a NAME="E11:126_2_1"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E12:126_2_1"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E13:126_2_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T17" target="_self">Th17</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E14:126_2_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T16" target="_self">Th16</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E15:126_2_1"/><b>then </b><i><font color="Green">E120</font></i>: 
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:126_2_1"/>
( 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T14" target="_self">Th14</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:126_2_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> &amp; 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, , <a class="ref" href="abcmiz_0.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:126_2_1"/><b>then </b><i><font color="Green">E121</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="abcmiz_0.html#D16" target="_self">Def16</a>, <a class="ref" href="abcmiz_0.html#T20" target="_self">Th20</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="abcmiz_0.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="abcmiz_0.html#T16" target="_self">Th16</a>, <a class="ref" href="abcmiz_0.html#D15" target="_self">Def15</a></i>
;<br/><b>hence </b><a NAME="E20:126_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="abcmiz_0.html#T14" target="_self">Th14</a>, <a class="ref" href="abcmiz_0.html#D15" target="_self">Def15</a>, <a class="ref" href="abcmiz_0.html#T21" target="_self">Th21</a>, <a class="ref" href="abcmiz_0.html#T22" target="_self">Th22</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:126_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E18:126"/><i><font color="Green">E122</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="abcmiz_0.html#T11" target="_self">Th11</a>, <a class="ref" href="abcmiz_0.html#T13" target="_self">Th13</a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:126_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:126_3"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E4:126_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:126_3"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> implies <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="abcmiz_0.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:126"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="finseq_5.html#K3">Rev</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font>
 
<b>by </b><i>, <a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E21:126"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_6.html#T29">FINSEQ_6:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<a NAME="E23:126"/><i><font color="Green">E179</font></i>: 
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a> <font color="Maroon">t2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D28" target="_self">ABCMIZ_0:def 28</a><br/><b>let </b><font color="Maroon">a</font> be   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:126_4"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/>

<a NAME="E2:126_4"/><b>then </b>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:126_4"/><i><font color="Green">E198</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:126_4"/><i><font color="Green">E199</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E7:126_4"/><b>then </b><i><font color="Green">E207</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D15" target="_self">Def15</a>, <a class="ref" href="abcmiz_0.html#D16" target="_self">Def16</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:126_4"/><b>then </b>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="abcmiz_0.html#D16" target="_self">Def16</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E9:126_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T20" target="_self">Th20</a>, <a class="ref" href="abcmiz_0.html#D16" target="_self">Def16</a></i>;<br/>
<b>then consider </b><font color="Maroon">o</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> being   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font>, <font color="Maroon">u</font> being   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E11:126_4"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">b</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">u</font> )
 <b>by </b><i/>;<br/>
<a NAME="E12:126_4"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">o</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="abcmiz_0.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E13:126_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">o</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">o</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="abcmiz_0.html#T21" target="_self">Th21</a>, <a class="ref" href="abcmiz_0.html#T22" target="_self">Th22</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E14:126_4"/><b>then </b><i><font color="Green">E211</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p4">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">o</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:126_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D15" target="_self">Def15</a>, <a class="ref" href="abcmiz_0.html#T22" target="_self">Th22</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:126_4"/>
<font color="Maroon">a</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T16" target="_self">Th16</a>, <a class="ref" href="abcmiz_0.html#D15" target="_self">Def15</a>, <a class="ref" href="abcmiz_0.html#T22" target="_self">Th22</a>, <a class="ref" href="abcmiz_0.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:126"/><i><font color="Green">E212</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/>
<b>thus </b><a NAME="E25:126"/>
<font color="Maroon">A</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">t2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_5"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D29" target="_self">ABCMIZ_0:def 29</a><br/>

<b>thus </b><a NAME="E1:126_5"/>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a> <font color="Maroon">t2</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E26:126"/>
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T17" target="_self">Th17</a>, </i>;<br/>
<b>then </b><font color="Maroon">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">t2</font> = 
<span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t2</font>
<b>by </b><i><a class="ref" href="abcmiz_0.html#D15" target="_self">Def15</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">t2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/>
<b>hence </b><a NAME="E28:126"/>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T10" target="_self">Th10</a>, <a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="abcmiz_0.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_5.html#T65">FINSEQ_5:65</a></i>;<br/>


</div>
