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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="waybel_1.html#V8" title="WAYBEL_1:attr.8">kernel</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">k</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  holds <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">k</font></span>)</span>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> holds <br/>( <font color="Olive" title="b2">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b3">y</font> iff <font color="Olive" title="b4">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b5">b</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">k</font> be   <a href="waybel_1.html#V8" title="WAYBEL_1:attr.8">kernel</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">k</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">k</font></span>)</span>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">y</font> iff <font color="Olive" title="b3">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b4">b</font> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">g</font> =  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">k</font>;<br/>
<span class="kw">assume </span><a NAME="E1:58"/>
<font color="Maroon" title="c2">k</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">k</font></span>)</span>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">y</font> iff <font color="Olive" title="b3">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b4">b</font> )</span><br/>

<a NAME="E2:58"/><span class="kw">then </span>
(  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">k</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  &amp;  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">k</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel34.html#T33" target="_self" title="WAYBEL34:th.33">Th33</a>, <a class="ref" href="waybel34.html#T34" target="_self" title="WAYBEL34:th.34">Th34</a></span>;<br/>
<a NAME="E3:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">k</font></span>)</span> is  <a href="waybel34.html#V1" title="WAYBEL34:attr.1">waybelow-preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel34.html#T24" target="_self" title="WAYBEL34:th.24">Th24</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">k</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> holds <br/>( <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">y</font> iff <font color="Olive" title="b1">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b2">b</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">k</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> implies ( <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">y</font> iff <font color="Maroon" title="c6">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c7">b</font> ) )</span><br/>



<a NAME="E4:58"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">k</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T56" title="WAYBEL_1:th.56">WAYBEL_1:56</a></span>;<br/>
<a NAME="E5:58"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">k</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel34.html#T33" target="_self" title="WAYBEL34:th.33">Th33</a></span>;<br/>
<a NAME="E6:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<span class="p1">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/>
<a NAME="E7:58"/>
<span class="p1">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:58"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> implies ( <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">y</font> iff <font color="Maroon" title="c6">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c7">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="waybel34.html#D8" target="_self" title="WAYBEL34:def.8">Def8</a>, <a class="ref" href="waybel34.html#T36" target="_self" title="WAYBEL34:th.36">Th36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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