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<b>set </b><font color="Maroon">I</font> =  <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">X</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">a<sub>1</sub></font>,<font color="Olive">r</font>,<font color="Olive">f</font> #) where <font color="Olive">r</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>, <font color="Olive">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">a<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> :  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">a<sub>1</sub></font>,<font color="Olive">r</font>,<font color="Olive">f</font> #) is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">X</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">M</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <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">X</font><b> such that </b><br/><a NAME="E2:69_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</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">X</font> holds <br/><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i>
<a NAME="E3:69_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>take </b><font color="Maroon">IT</font> =  <a href="card_3.html#K3">Union</a> <font color="Maroon">M</font>;<br/>

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:69_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 ;<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="E3:69_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E4:69_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T5" target="_self">Th5</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:69_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E7:69_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E4:69_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:69_1_1_1"/>
<font color="Maroon">z</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">X</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T6" target="_self">Th6</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:69_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">z</font>,<font color="Olive">r</font>,<font color="Olive">f</font> #) where <font color="Olive">r</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">z</font>,<font color="Maroon">z</font>, <font color="Olive">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">z</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> :  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">z</font>,<font color="Olive">r</font>,<font color="Olive">f</font> #) is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">X</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">r</font> being    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">z</font>,<font color="Maroon">z</font>, <font color="Maroon">f</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">z</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E11:69_1_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">z</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #)
 <b>and </b><br/><a NAME="E12:69_1_1_1"/><i><font color="Green">E118</font></i>: 
 <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">z</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #) is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:69_1_1_1"><i><font color="Green">E41</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">N</font> =  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">z</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #) as  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">N</font> = <font color="Maroon">N</font>;<br/><b>thus </b><a NAME="E14:69_1_1_1"/>
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E15:69_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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">X</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T6" target="_self">Th6</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">N</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E5:69_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:69_1_1"/><i><font color="Green">E127</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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">X</font>
 ;<br/>

<b>set </b><font color="Maroon">i</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>;<br/>
<a NAME="E7:69_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:69_1_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:69_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,<font color="Olive">r</font>,<font color="Olive">f</font> #) where <font color="Olive">r</font> is    <a href="relset_1.html#M2">Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>, <font color="Olive">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> :  <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,<font color="Olive">r</font>,<font color="Olive">f</font> #) is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:69_1_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E11:69_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:69_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T5" target="_self">Th5</a>, , <a class="ref" href="yellow_6.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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