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<b>let </b><font color="Maroon">U0</font> be  <a href="unialg_2.html#V2">with_const_op</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">H</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="unialg_3.html#K1" target="_self">Sub</a> <font color="Maroon">U0</font></span>)</span>;<br/>



<b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">U0</font>;<br/>
<a NAME="E2:20"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">S</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" target="_self">Carr</a> <font color="Maroon">U0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" target="_self">Carr</a> <font color="Maroon">U0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font> ;<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="unialg_3.html#M2" target="_self">Element</a> of  <a href="unialg_3.html#K1" target="_self">Sub</a> <font color="Maroon">U0</font><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" target="_self">Carr</a> <font color="Maroon">U0</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X1</font> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<a NAME="E6:20_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:20_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">CH</font> = <span class="p1">(<span class="default"><a href="unialg_3.html#K6" target="_self">Carr</a> <font color="Maroon">U0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">U0</font> ;<br/>
<a NAME="E4:20"/>
<font color="Maroon">CH</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="E5:20"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">CH</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:20"/>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_3.html#K6" target="_self">Carr</a> <font color="Maroon">U0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font></span>)</span> is  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font>
 ;<br/>


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