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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> , <font color="Maroon">T</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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:30"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><a NAME="E2:30"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">c</font> ) holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D7">YELLOW_0:def 7</a><br/>

<a NAME="E5:30"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a>  is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:30"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E7:30"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D7">YELLOW_0:def 7</a><br/>

<b>thus </b><a NAME="E8:30"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="waybel13.html#T19">WAYBEL13:19</a></i>;<br/>

<a NAME="E9:30"/><i><font color="Green">E76</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:30"/><i><font color="Green">E77</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">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1"/>
<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">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</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>thus </b><a NAME="E2:30_1"/>
<font color="Maroon">A</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">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>


</div><b>end;</b></div>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:30_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:30_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <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">A</font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="waybel13.html#T19">WAYBEL13:19</a></i>;<br/><a NAME="E3:30_2"/><b>then </b>
<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">A</font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T68">TOPS_2:68</a></i>;<br/><a NAME="E4:30_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:30_2"/><b>then </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E6:30_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E12:30"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E13:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <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">A</font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="waybel13.html#T19">WAYBEL13:19</a></i>;<br/>
<a NAME="E14:30"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T68">TOPS_2:68</a></i>;<br/>
<b>assume </b><a NAME="E15:30"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E16:30"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:30_3"/>
<font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:30_3"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="waybel13.html#T19">WAYBEL13:19</a></i>;<br/>
<a NAME="E3:30_3"/><b>then </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:30_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<b>hence </b><a NAME="E5:30_3"/>
<font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E18:30"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>


</div>
