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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> non <a href="abcmiz_0.html#V4" target="_self">void</a>  <a href="abcmiz_0.html#L3" target="_self">TAS-structure</a> ;<br/><b>let </b><font color="Maroon">t</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">v1</font> be    <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>, <font color="Maroon">v2</font> be    <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>;<br/>





<b>set </b><font color="Maroon">v</font> = <font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font>;<br/>
<b>assume </b><a NAME="E1:104"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">a</font> being  <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font><br/> for <font color="Olive">s</font> being  <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">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"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Olive">a</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Olive">s</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D28" target="_self">ABCMIZ_0:def 28</a><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D28" target="_self">ABCMIZ_0:def 28</a>
<div class="add"><b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</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="E1:104_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/><a NAME="E2:104_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/><a NAME="E3:104_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E4:104_1"/>
<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#D6" target="_self">Def6</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">i</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="E3:104"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E4:104"/><i><font color="Green">E46</font></i>: 
 <a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:104"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:104"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E8:104"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:104"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &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> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &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">v2</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T6" target="_self">Th6</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:104"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <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="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</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> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E11:104"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</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> <font color="Maroon">v1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> &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"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>,<font color="Maroon">t</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> <font color="Maroon">v1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T4" target="_self">Th4</a>, <a class="ref" href="abcmiz_0.html#T5" target="_self">Th5</a>, <a class="ref" href="abcmiz_0.html#T6" target="_self">Th6</a>, , <a class="ref" href="finseq_1.html#T41">FINSEQ_1:41</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E12:104"/>
<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#D6" target="_self">Def6</a></i>;<br/>


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