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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">J</font> be    <a href="yellow_6.html#M3" target="_self">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:82"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font></span>)</span> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:82"/><b>then </b><i><font color="Green">E41</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E5:82"/><i><font color="Green">E42</font></i>: 
 ex <font color="Olive">N</font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:82"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:82"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:82_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:82_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 <b>and </b><br/><a NAME="E4:82_1"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="yellow_6.html#D4" target="_self">Def4</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:82_1"/>
<font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:82_1"/><b>then </b>
 ex <font color="Olive">N</font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:82_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:82"/><b>then </b>
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:82"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E10:82"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self">Th9</a>, , <a class="ref" href="classes2.html#T67">CLASSES2:67</a></i>;<br/>
<b>hence </b><a NAME="E11:82"/>
 <a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


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