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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">C</font> being   <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E2:46_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font> holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon">M</font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font>;<br/>
<a NAME="E3:46_1_1"/>
<font color="Maroon">B</font> is    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:46_1_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp;  ex <font color="Olive">C</font> being   <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> ) )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>
<a NAME="E5:46_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:46_1_1"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp;  ex <font color="Olive">C</font> being   <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> ) ) )
 
<b>by </b><i/>;<br/>
<a NAME="E7:46_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> is <a href="polyalg1.html#V3" target="_self">opers_closed</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">C</font> being  <a href="polyalg1.html#V1" target="_self">strict</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E9:46_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:46_1_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as  non <a href="struct_0.html#V3">empty</a> <a href="polyalg1.html#V1" target="_self">strict</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:46_1_1_1"/>
 ex <font color="Olive">C</font> being   <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:46_1_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:46_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>

<b>let </b><font color="Maroon">C1</font> be    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E14:46_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>
 ;<br/>

<a NAME="E15:46_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:46_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E17:46_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>


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