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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>   <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a>  non  <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>   <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a>   <a href="abcmiz_0.html#L2" title="ABCMIZ_0:struct.2">TA-structure</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Olive" title="b1">t</font> holds <br/>for <font color="Olive" title="b3">s</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">t</font></span>)</span> holds <br/>( <font color="Olive" title="b2">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Olive" title="b1">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s</font> &amp; <font color="Olive" title="b3">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">t</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c2">t</font> holds <br/>for <font color="Olive" title="b2">s</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">t</font></span>)</span> holds <br/>( <font color="Olive" title="b1">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">t</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c2">t</font> implies for <font color="Olive" title="b1">s</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c3">v</font>,<font color="Maroon" title="c2">t</font></span>)</span> holds <br/>( <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">t</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c2">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c3">v</font>,<font color="Maroon" title="c2">t</font></span>)</span> holds <br/>( <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">t</font> )</span><br/>

<a NAME="E2:78"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v</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="c3">v</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c3">v</font>,<font color="Maroon" title="c2">t</font></span>)</span> implies ( <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c4">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">t</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:78"/>
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c3">v</font>,<font color="Maroon" title="c2">t</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c4">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">t</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:78"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">t</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:78"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<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="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v</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">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E45">A3</font></span></a></span>;<br/>
<a NAME="E8:78"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E9:78"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>;<br/>
<a NAME="E10:78"/>
<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:78"/>
( <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c4">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E9:78"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="abcmiz_0.html#T42" target="_self" title="ABCMIZ_0:th.42">Th42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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