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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="abcmiz_0.html#L2" target="_self">TA-structure</a> ;<br/>

<b>assume </b><a NAME="E1:30"/><i><font color="Green">E12</font></i>: 
 <a href="abcmiz_0.html#G2" target="_self">TA-structure</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="abcmiz_0.html#U2" target="_self">non-op</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="abcmiz_0.html#G2" target="_self">TA-structure</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="abcmiz_0.html#U2" target="_self">non-op</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">c<sub>2</sub></font> #)
 ;<br/>

<b>assume </b><a NAME="E2:30"/>
the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">c<sub>1</sub></font> is  <a href="waybel_0.html#V20">join-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>3</sub></font> = the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">c<sub>1</sub></font> as  <a href="waybel_0.html#V20">join-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:30"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green">E13</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) &amp;  <a href="orders_2.html#G1">RelStr</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">c<sub>1</sub></font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>,the <a href="orders_2.html#U1">InternalRel</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">c<sub>1</sub></font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</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">c<sub>2</sub></font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>,the <a href="orders_2.html#U1">InternalRel</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">c<sub>2</sub></font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> #) )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E1:30"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E6:30"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V20">join-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:30"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:30_1"/><i><font color="Green">E14</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<a NAME="E3:30_1"/><b>then </b><i><font color="Green">E15</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E5:30"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a>, <a class="ref" href="yellow_0.html#T14">YELLOW_0:14</a></i>;<br/>
<a NAME="E4:30_1"/><b>then </b><i><font color="Green">E16</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span>,<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">c<sub>1</sub></font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a>  &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<b>thus </b><a NAME="E5:30_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span>,<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">c<sub>2</sub></font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> 
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:30"><i><font color="Green">E12</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:30_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>

<b>thus </b><a NAME="E6:30_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E5:30"><i><font color="Green">E13</font></i></a>, <a class="txt" href="abcmiz_0.html#E2:30_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="abcmiz_0.html#E4:30_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:30"/>
the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">c<sub>2</sub></font> is  <a href="waybel_0.html#V20">join-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:30"><i><font color="Green">E12</font></i></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>
