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<div class="add">

<b>set </b><font color="Maroon" title="c2">IT</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> ) </span>}</span> ;<br/>
<a NAME="E1:16_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> ) </span>}</span> 
 ;<br/>

<a NAME="E2:16_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> )
 
;<br/>
<b>hence </b><a NAME="E3:16_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">IT</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b1">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<a NAME="E3:16_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">IT</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>
 
;<br/>
<b>hence </b><a NAME="E4:16_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">S</font> where <font color="Olive" title="b2">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> : ( <font color="Olive" title="b2">S</font> is <a href="wellfnd1.html#V2" title="WELLFND1:attr.2">well_founded</a> &amp; <font color="Olive" title="b2">S</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> ) </span>}</span> 
 ;<br/>


</div>
