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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> <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">R</font> = <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> ;<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>assume </b><a NAME="E1:133"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">t1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t2</font>
 ;<br/>

<a NAME="E2:133"/>
<font color="Maroon">t2</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:133"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">t1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">t1</font>, <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t1</font> )
 
<b>by </b><i>, <a class="ref" href="abcmiz_0.html#D12" target="_self">Def12</a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</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">T</font>,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>] <b>means </b>( <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">t2</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">t2</font> );<br/>
<div><i><font color="Green">E43</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">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/><b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">x</font>, <font color="Maroon">t2</font> = <font color="Maroon">y</font> as   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</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">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> 
 ;<br/><a NAME="E3:133_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font> st <br/>( not <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t2</font> &amp; <font color="Olive">a</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">t2</font> &amp; <font color="Olive">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:133_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E5:133"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">x</font>]
 
;<br/>
<a NAME="E6:133"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
;<br/>
<a NAME="E7:133"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Olive">x</font>,<font color="Olive">y</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<b>from </b><i><a class="ref" href="abcmiz_0.html#S2" target="_self">ABCMIZ_0:sch 2</a>(<a class="ref" href="abcmiz_0.html#T11" target="_self">Th11</a>, <a class="ref" href="abcmiz_0.html#T13" target="_self">Th13</a>, );<br/></i>
<b>hence </b><a NAME="E8:133"/>
 <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a></i>;<br/>


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