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<b>let </b><font color="Maroon">S</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">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">X</font> be  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>set </b><font color="Maroon">f</font> = <font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font>;<br/>
<a NAME="E1:63"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font> <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> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D2">BORSUK_1:def 2</a></i>;<br/>
<a NAME="E2:63"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font></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"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_1"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font></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"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T6">BORSUK_1:6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E3:63_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E4:63_1"/><b>then </b>
<font color="Maroon">t</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> <font color="Maroon">t</font></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 NAME="E5:63_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:63"/>
( <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font></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> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<b>hence </b><a NAME="E4:63"/>
( <font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">t</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>


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