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<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">T1</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">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="E1:9"/><i><font color="Green">E26</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">S2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S2</font> #) &amp;  <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> #) <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">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">T1</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">T2</font>;<br/>



<b>assume </b><a NAME="E2:9"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f2</font>
 ;<br/>

<b>thus </b><a NAME="E3:9"/>
( <font color="Maroon">f1</font> is <a href="waybel_0.html#V17">infs-preserving</a> implies <font color="Maroon">f2</font> is <a href="waybel_0.html#V17">infs-preserving</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font> holds  <font color="Maroon">f1</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a><br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a><br/>

<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font> <b>by </b><i/>;<br/>
<a NAME="E3:9_1"/>
<font color="Maroon">f1</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:9_1"/>
<font color="Maroon">f2</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#T65">WAYBEL_0:65</a></i>;<br/>


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

<b>assume </b><a NAME="E4:9"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font>  st not <font color="Olive">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">X</font> is <a href="waybel_0.html#V1">directed</a> holds <br/><font color="Maroon">f1</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a><br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a><br/>

<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E6:9"/>
( not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">X</font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<a NAME="E7:9"/><b>then </b>
( not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y</font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T3">WAYBEL_0:3</a></i>;<br/>
<a NAME="E8:9"/><b>then </b>
<font color="Maroon">f1</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:9"/>
<font color="Maroon">f2</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T65">WAYBEL_0:65</a></i>;<br/>


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