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<b>let </b><font color="Maroon">c<sub>1</sub></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> <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#V14" target="_self">commutative</a>  <a href="abcmiz_0.html#L3" target="_self">TAS-structure</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> ;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:133"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:133"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T80" target="_self">Th80</a></i>;<br/>
<a NAME="E3:133"/><b>then </b><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>, <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E1:133"><i><font color="Green">E85</font></i></a>, <a class="ref" href="abcmiz_0.html#T79" target="_self">Th79</a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>4</sub></font> );<br/>
<div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>assume </b><a NAME="E2:133_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> 
 ;<br/><a NAME="E3:133_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D31" target="_self">Def31</a></i>;<br/><b>hence </b><a NAME="E4:133_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T82" target="_self">Th82</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:133"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<a NAME="E6:133"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
;<br/>
<a NAME="E7:133"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<b>from </b><i><a class="ref" href="abcmiz_0.html#S2" target="_self">ABCMIZ_0:sch 2</a>(<a class="txt" href="abcmiz_0.html#E4:133"><i><font color="Green">E87</font></i></a>, <a class="txt" href="abcmiz_0.html#E5:133"><i><font color="Green">E88</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:133"><i><font color="Green">E89</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:133"/>
 <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:133"><i><font color="Green">E86</font></i></a></i>;<br/>


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