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<div class="add">

<b>thus </b><a NAME="E1:31_1_1"/>
( <font color="Maroon">T</font> is <a href="abcmiz_0.html#V9" target="_self">adj-structured</a> implies for <font color="Olive">t1</font>, <font color="Olive">t2</font> being  <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> holds   <a href="abcmiz_0.html#K2" target="_self">adjs</a> <span class="p1">(<span class="default"><font color="Olive">t1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">t2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Olive">t1</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Olive">t2</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1_1"/>
the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">T</font> is  <a href="waybel_0.html#V20">join-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D10" target="_self">ABCMIZ_0:def 10</a><br/>

<b>then reconsider </b><font color="Maroon">f</font> = the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">T</font> as  <a href="waybel_0.html#V20">join-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> ;<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>thus </b> <a href="abcmiz_0.html#K2" target="_self">adjs</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">t2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_6.html#T2">WAYBEL_6:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_7.html#T22">YELLOW_7:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t1</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t2</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:31_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">t1</font>, <font color="Olive">t2</font> being  <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> holds   <a href="abcmiz_0.html#K2" target="_self">adjs</a> <span class="p1">(<span class="default"><font color="Olive">t1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">t2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Olive">t1</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Olive">t2</font></span>)</span>
 ;<br/>

 <a href="yellow_1.html#K3">BoolePoset</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> = 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>
<br/>.= 
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span></span>)</span> #)
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>
;<br/>
<a NAME="E4:31_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a>  <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span></span>)</span> <a href="lattice3.html#K6">~</a> </span>)</span> #)
 
;<br/>
<a NAME="E5:31_1_1"/>
(  <a href="relset_1.html#K5">rng</a> the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="finsub_1.html#K5">Fin</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> &amp;  <a href="finsub_1.html#K5">Fin</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> 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="finsub_1.html#T26">FINSUB_1:26</a></i>;<br/>
<a NAME="E6:31_1_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">T</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_3"><b>now </b></a><div class="add"><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>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">t2</font></span>)</span> = 
 <a href="abcmiz_0.html#K2" target="_self">adjs</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">t2</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t1</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_7.html#T22">YELLOW_7:22</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:31_1_1"/>
the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">T</font> is  <a href="waybel_0.html#V20">join-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel_6.html#T2">WAYBEL_6:2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D10" target="_self">ABCMIZ_0:def 10</a><br/>


</div>
