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

<b>set </b><font color="Maroon">t</font> =  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">f</font> = <font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span>;<br/>
<b>take </b>
<font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span>
;<br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a><br/>

<b>assume </b><a NAME="E1:9_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E2:9_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:9_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>
<a NAME="E4:9_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#R2">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D2">BORSUK_1:def 2</a></i>;<br/>
<a NAME="E5:9_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:9_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:9_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:9_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/><a NAME="E3:9_1_1_1_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D11">YELLOW_0:def 11</a></i>;<br/><b>hence </b><a NAME="E4:9_1_1_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X1</font>;<br/><a NAME="E4:9_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><b>hence </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> = 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
