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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> non <a href="abcmiz_0.html#V4" target="_self">void</a>  <a href="abcmiz_0.html#L3" target="_self">TAS-structure</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> );<br/>
<b>assume </b><a NAME="E1:110"/>
<font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:110"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T63" target="_self">Th63</a></i>;<br/>
<a NAME="E3:110"/><b>then </b><i><font color="Green">E66</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:110"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>and </b><br/><a NAME="E6:110"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>from </b><i><a class="ref" href="abcmiz_0.html#S1" target="_self">ABCMIZ_0:sch 1</a>(<a class="txt" href="abcmiz_0.html#E3:110"><i><font color="Green">E66</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="abcmiz_0.html#E5:110"><i><font color="Green">E67</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E8:110"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E5:110"><i><font color="Green">E67</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E9:110"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E10:110"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E5:110"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E11:110"/>
(  not <font color="Maroon">c<sub>6</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:110"><i><font color="Green">E68</font></i></a>, <a class="txt" href="abcmiz_0.html#E9:110"><i><font color="Green">E69</font></i></a></i>;<br/>


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