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<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>  implies ( <font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:124_2_1"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
 for <font color="Olive" title="b1">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font> holds  <font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">X</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D33" target="_self" title="WAYBEL_0:def.33">WAYBEL_0:def 33</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E2:124_2_1"/>
 for <font color="Olive" title="b1">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font>  st  not <font color="Olive" title="b1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive" title="b1">X</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  holds <br/><font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">X</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D37" target="_self" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:124_2_1"/>
 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">L1</font> holds  <font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:124_2_1"><span class="lab"><font color="Green" title="E55">A2</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D35" target="_self" title="WAYBEL_0:def.35">WAYBEL_0:def 35</a> <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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