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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E1:22"/><i><font color="Green">E15</font></i>: 
 <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="wellfnd1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E2:22"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:22"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>assume </b><a NAME="E5:22"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E6:22"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V12">lower</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E7:22"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> is  <a href="wellfnd1.html#V2" target="_self">well_founded</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E8:22"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E9:22"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="wellfnd1.html#V2" target="_self">well_founded</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E6:22"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:22"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:22"><i><font color="Green">E15</font></i></a>, <a class="txt" href="wellfnd1.html#E3:22"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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