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

<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>given </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font><b> such that </b><a NAME="E1:24"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a><br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; ( for <font color="Olive">c'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> holds <br/><font color="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a><br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a><br/>

<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:24"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</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">x</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</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">y</font> )
 <b>and </b><br/><a NAME="E5:24"/><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="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</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">x</font> &amp; <font color="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</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">y</font> holds <br/><font color="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:24"/><i><font color="Green">E44</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> the <a href="struct_0.html#U1">carrier</a> of <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="E7:24"/><i><font color="Green">E45</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">c</font>
;<br/>

<a NAME="E8:24"/>
( <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <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">x</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <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">y</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<b>hence </b><a NAME="E9:24"/>
( <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>

<b>let </b><font color="Maroon">z'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E10:24"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E11:24"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  &amp; <font color="Olive">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="E12:24"/><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">z'</font> <a href="orders_2.html#R1">&lt;=</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">x</font> &amp; <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">z'</font> <a href="orders_2.html#R1">&lt;=</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">y</font> )
 
<b>by </b><i>, , <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>
<a NAME="E13:24"/><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">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:24"/><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">z'</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <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="E15:24"/>
<font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>


</div>
