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<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="lattice5.html#NM1">Homomorphism</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font></span>)</span>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E58</font></i>: 
not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:14_1"/><i><font color="Green">E59</font></i>: 
(  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="waybel_1.html#K2">corestr</a> <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_1.html#T32">WAYBEL_1:32</a></i>;<br/><a NAME="E3:14_1"/><i><font color="Green">E60</font></i>: 
(  <a href="lattice5.html#K2">Image</a> <font color="Maroon">f</font> is   <a href="waybel_0.html#NM6">Semilattice</a> &amp; <font color="Maroon">L</font> is   <a href="waybel_0.html#NM6">Semilattice</a> )
 ;<br/><a NAME="E4:14_1"/><i><font color="Green">E61</font></i>: 
 <a href="lattice5.html#K2">Image</a> <font color="Maroon">f</font> is  <a href="orders_2.html#V1">strict</a> <a href="yellow_0.html#V4">full</a> <a href="lattice5.html#NM2">Sublattice</a> of  <a href="lattice5.html#K1">EqRelLATT</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E5:14_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

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

<b>thus </b><span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> = 
<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#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_6.html#T1">WAYBEL_6:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T70">YELLOW_0:70</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E6:14_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/><a NAME="E7:14_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E8:14_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D1">WAYBEL_1:def 1</a></i>;<br/><b>hence </b><a NAME="E9:14_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>


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