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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L</font> holds <br/> (  <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">B</font></span>)</span> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp;  <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">B</font></span>)</span> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font> be   <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>   <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp;  <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>  )</span><br/>



<a NAME="E1:105"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
 <a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel13.html#T33" title="WAYBEL13:th.33">WAYBEL13:33</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:105"/>
 <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> 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="waybel23.html#T64" target="_self" title="WAYBEL23:th.64">Th64</a>, <a class="ref" href="waybel_1.html#T12" title="WAYBEL_1:th.12">WAYBEL_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> </span><br/>

<a NAME="E3:105"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
 <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#T62" target="_self" title="WAYBEL23:th.62">Th62</a></span>;<br/>
<a NAME="E4:105"/>
 <a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#T61" target="_self" title="WAYBEL23:th.61">Th61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:105"/>
 <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="waybel20.html#T27" title="WAYBEL20:th.27">WAYBEL20:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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