<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="pralg_1.html#V2">1-sorted-yielding</a> <a href="waybel_3.html#V4">non-Empty</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>





<a NAME="E1:39"/>
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E3:39_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:39_1"/><b>then </b>
<font color="Maroon">A</font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:39_1"/><b>then </b>
not <font color="Maroon">A</font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">j</font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D4">YELLOW_6:def 4</a></i>;<br/>
<a NAME="E6:39_1"/><b>then </b><i><font color="Green">E35</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">j</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E7:39_1"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D17">PBOOLE:def 17</a></i>;<br/>
<a NAME="E8:39_1"/><b>then </b>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/>
<a NAME="E9:39_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:39_1"/><b>then </b>
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:39_1"/>
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:39"/>
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="pzfmisc1.html#K1">}</a></span> is    <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>


</div>
