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<b>let </b><font color="Maroon">L1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">L2</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">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font>;<br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<a NAME="E2:19"/><b>then </b>
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E3:19"/>
( <font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="yellow_2.html#T15">YELLOW_2:15</a></i>;<br/>

<a NAME="E4:19"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<b>thus </b><a NAME="E5:19"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> implies <font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L2</font>,<font color="Maroon">L1</font> <b>by </b><i>, <a class="ref" href="waybel_0.html#T67">WAYBEL_0:67</a></i>;<br/>
<a NAME="E2:19_1"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T68">WAYBEL_0:68</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<a NAME="E4:19_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</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="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/>
<a NAME="E5:19_1"/>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> )
 
;<br/>
<a NAME="E6:19_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:19_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</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>
 
<b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<i><font color="Green">E50</font></i>: <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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> = 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</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>
<b>by </b><i>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/>
<a NAME="E9:19_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> &amp; not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font> is <a href="xboole_0.html#V1">empty</a> )
 
;<br/>
<a NAME="E10:19_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E11:19_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E12:19_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T15">YELLOW_2:15</a></i>;<br/>
<b>hence </b><a NAME="E13:19_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>


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


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