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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>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>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:85"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</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>
 ;<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="E3:85"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</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>
 <b>and </b><br/><a NAME="E4:85"/><i><font color="Green">E32</font></i>: 
the <a href="waybel_0.html#U1">mapping</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="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/>
<a NAME="E5:85"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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>
 
<b>by </b><i/>;<br/>
<a NAME="E6:85"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <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><b> such that </b><br/><a NAME="E8:85"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E9:85"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
;<br/>
<a NAME="E10:85"/>
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <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>
 
;<br/>
<a NAME="E11:85"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> 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>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E12:85"/>
<font color="Maroon">y1</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>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E13:85"/><b>then </b>
<font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <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">y1</font>
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E14:85"/><b>then </b>
<font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">y1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:85"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">y1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon">x</font> = 
the <a href="waybel_0.html#U1">mapping</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="binop_1.html#K1">.</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
<b>by </b><i>, <a class="ref" href="yellow_6.html#D4" target="_self">Def4</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">y1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E17:85"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">y1</font></span>)</span>
 
<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">y1</font> = <font color="Maroon">y1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E19:85"/>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">y1</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E20:85"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E11:85"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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