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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_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</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">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  holds <br/> <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">p</font> is   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> )</span><br/>



<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">Lk</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">k</font> </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T43" onmouseover="rs('waybel_1/T43')" onmouseout="rh()">WAYBEL_1:43</a></span>;<br/>
<a NAME="E3:17"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is  <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T50" onmouseover="rs('waybel_1/T50')" onmouseout="rh()">WAYBEL_1:50</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">pk</font> = <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Lk</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T46" onmouseover="rs('waybel_1/T46')" onmouseout="rh()">WAYBEL_1:46</a></span>;<br/>
<a NAME="E5:17"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c4">pk</font> is  <a href="waybel_1.html#V8" title="WAYBEL_1:attr.8">kernel</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T48" onmouseover="rs('waybel_1/T48')" onmouseout="rh()">WAYBEL_1:48</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>  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="c4">pk</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">X</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c5">X</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  implies <font color="Maroon" title="c4">pk</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c5">X</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">X9</font> = <font color="Maroon" title="c5">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel13.html#T3" onmouseover="rs('waybel13/T3')" onmouseout="rh()">WAYBEL13:3</a></span>;<br/><span class="kw">assume </span><a NAME="E2:17_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
(  not <font color="Maroon" title="c5">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c5">X</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">pk</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c5">X</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">X9</font> = <font color="Maroon" title="c6">X9</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T7" onmouseover="rs('yellow_2/T7')" onmouseout="rh()">YELLOW_2:7</a></span>;<br/><a NAME="E4:17_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c7">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="waybel_0.html#D37" onmouseover="rs('waybel_0/D37')" onmouseout="rh()">WAYBEL_0:def 37</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> implies (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">X</font></span>)</span> ) )</span></a><div class="add"><a NAME="E1:17_1_1"/>
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>
 ;<br/><a NAME="E2:17_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Lk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>;<br/><a NAME="E3:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T129" onmouseover="rs('relat_1/T129')" onmouseout="rh()">RELAT_1:129</a></span>;<br/><span class="kw">assume </span><a NAME="E4:17_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">X</font></span>)</span> )</span><br/><a NAME="E5:17_1_1"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is  <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T50" onmouseover="rs('waybel_1/T50')" onmouseout="rh()">WAYBEL_1:50</a></span>;<br/><a NAME="E6:17_1_1"/><span class="kw">then </span>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T30" onmouseover="rs('yellow_2/T30')" onmouseout="rh()">YELLOW_2:30</a></span>;<br/><span class="kw">hence </span><a NAME="E7:17_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" onmouseover="rs('yellow_0/T17')" onmouseout="rh()">YELLOW_0:17</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">X</font></span>)</span></span><br/><a NAME="E8:17_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" onmouseover="rs('yellow_0/T17')" onmouseout="rh()">YELLOW_0:17</a></span>;<br/><a NAME="E9:17_1_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="waybel_1.html#T52" onmouseover="rs('waybel_1/T52')" onmouseout="rh()">WAYBEL_1:52</a></span>;<br/><a NAME="E10:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A9</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">pk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">X9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E8:17_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="waybel_0.html#T7" onmouseover="rs('waybel_0/T7')" onmouseout="rh()">WAYBEL_0:7</a></span>;<br/><a NAME="E11:17_1_1"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font> is   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="yellow_0.html#T17" onmouseover="rs('yellow_0/T17')" onmouseout="rh()">YELLOW_0:17</a>, <a class="ref" href="yellow_2.html#T15" onmouseover="rs('yellow_2/T15')" onmouseout="rh()">YELLOW_2:15</a></span>;<br/><span class="kw">hence </span> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pk</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font></span>)</span> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E9:17_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E3:17_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="waybel_0.html#T7" onmouseover="rs('waybel_0/T7')" onmouseout="rh()">WAYBEL_0:7</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">X9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E8:17_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="waybel_0.html#D31" onmouseover="rs('waybel_0/D31')" onmouseout="rh()">WAYBEL_0:def 31</a></span>
<br/>.= 
<font color="Maroon" title="c4">pk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:17_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:17_1"/>
<font color="Maroon" title="c4">pk</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D31" onmouseover="rs('waybel_0/D31')" onmouseout="rh()">WAYBEL_0:def 31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c4">pk</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D37" onmouseover="rs('waybel_0/D37')" onmouseout="rh()">WAYBEL_0:def 37</a></span>;<br/>
<a NAME="E8:17"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="waybel_1.html#T52" onmouseover="rs('waybel_1/T52')" onmouseout="rh()">WAYBEL_1:52</a></span>;<br/>
<a NAME="E9:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A11</font></span>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="waybel_5.html#T28" onmouseover="rs('waybel_5/T28')" onmouseout="rh()">WAYBEL_5:28</a></span>;<br/>
<span class="lab"><font color="Green" title="E21">A12</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>
<br/>.= 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">pk</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T44" onmouseover="rs('waybel_1/T44')" onmouseout="rh()">WAYBEL_1:44</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">pk</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>
;<br/>
<a NAME="E11:17"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">pk</font></span>)</span> is   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel15.html#T1" target="_self" onmouseover="rs('waybel15/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E12:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A13</font></span>: 
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">pk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:17"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="yellow_0.html#T57" onmouseover="rs('yellow_0/T57')" onmouseout="rh()">YELLOW_0:57</a></span>;<br/>
<a NAME="E13:17"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="yellow_2.html#T30" onmouseover="rs('yellow_2/T30')" onmouseout="rh()">YELLOW_2:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:17"/>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="ref" href="waybel15.html#T14" target="_self" onmouseover="rs('waybel15/T14')" onmouseout="rh()">Th14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
