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

<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:122"/><i><font color="Green">E28</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S1</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T1</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S2</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T2</font> #) )
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">T2</font>;<br/>



<b>assume </b><a NAME="E2:122"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>;<br/>



<b>assume </b><a NAME="E3:122"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>thus </b><a NAME="E4:122"/>
( <font color="Maroon">f</font> <a href="waybel_0.html#R4" target="_self">preserves_sup_of</a> <font color="Maroon">X</font> implies <font color="Maroon">g</font> <a href="waybel_0.html#R4" target="_self">preserves_sup_of</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:122_1"/><i><font color="Green">E32</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">S1</font> implies (  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">S2</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <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> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31" target="_self">WAYBEL_0:def 31</a><br/>

<b>assume </b><a NAME="E2:122_1"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y</font>,<font color="Maroon">T1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31" target="_self">WAYBEL_0:def 31</a><br/>

<b>hence </b><a NAME="E3:122_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>,<font color="Maroon">T2</font>
 <b>by </b><i>, , , , <a class="ref" href="yellow_0.html#T14">YELLOW_0:14</a></i>;<br/>

<a NAME="E4:122_1"/>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <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#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , , , <a class="ref" href="waybel_0.html#D6" target="_self">Def6</a>, <a class="ref" href="yellow_0.html#T14">YELLOW_0:14</a>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<b>hence </b><a NAME="E5:122_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <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">Y</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="waybel_0.html#D6" target="_self">Def6</a>, <a class="ref" href="yellow_0.html#T14">YELLOW_0:14</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:122"/><i><font color="Green">E35</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">S1</font> implies (  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">S2</font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <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> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30" target="_self">WAYBEL_0:def 30</a><br/>

<b>assume </b><a NAME="E6:122"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>,<font color="Maroon">T1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30" target="_self">WAYBEL_0:def 30</a><br/>

<b>hence </b><a NAME="E7:122"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>,<font color="Maroon">T2</font>
 <b>by </b><i>, , , , <a class="ref" href="yellow_0.html#T14">YELLOW_0:14</a></i>;<br/>

<a NAME="E8:122"/>
(  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <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#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , , , <a class="ref" href="waybel_0.html#D6" target="_self">Def6</a>, <a class="ref" href="yellow_0.html#T14">YELLOW_0:14</a>, <a class="ref" href="yellow_0.html#T27">YELLOW_0:27</a></i>;<br/>
<b>hence </b><a NAME="E9:122"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <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">Y</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="waybel_0.html#D6" target="_self">Def6</a>, <a class="ref" href="yellow_0.html#T14">YELLOW_0:14</a></i>;<br/>


</div>
