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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_8.html#V2" title="WAYBEL_8:attr.2">algebraic</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</font> being   <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</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">c</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  holds <br/><font color="Olive" title="b1">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">c</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font> be   <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">c</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:47"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">c</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:47_1"/>
<font color="Maroon" title="c2">c</font> is  <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D13" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a></span>;<br/><a NAME="E2:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T19" title="YELLOW_2:th.19">YELLOW_2:19</a></span>;<br/><a NAME="E3:47_1"/>
<font color="Maroon" title="c2">c</font> is  <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D13" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">Imc</font> =  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> as   <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="txt" href="#E1:47"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="yellow_2.html#T35" title="YELLOW_2:th.35">YELLOW_2:35</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c4">a9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> implies <font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E5:47_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
 ; <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="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span><br/><a NAME="E6:47_1"/><span class="lab"><font color="Green" title="E24">A4</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="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c3">Imc</font></span>)</span> <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 <font color="Maroon" title="c3">Imc</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">a9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Imc</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>;<br/><a NAME="E8:47_1"/>
<font color="Maroon" title="c5">a</font> is  <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></span>;<br/><a NAME="E9:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></span>;<br/><a NAME="E10:47_1"/>
<font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Imc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E6:47_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>;<br/><a NAME="E11:47_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E13:47_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">and </span><br/><a NAME="E14:47_1"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c6">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a></span>;<br/><a NAME="E15:47_1"/>
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font> is   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="waybel_8.html#D4" target="_self" title="WAYBEL_8:def.4">Def4</a></span>;<br/><a NAME="E16:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A8</font></span>: 
( <font color="Maroon" title="c2">c</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c1">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></span>;<br/><a NAME="E17:47_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c2">c</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="yellow_2.html#T16" title="YELLOW_2:th.16">YELLOW_2:16</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:47_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span>
 ; <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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:47_1_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">and </span><br/><a NAME="E4:47_1_1"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">and </span><br/><a NAME="E5:47_1_1"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">v9</font> = <font color="Maroon" title="c8">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1"><span class="lab"><font color="Green" title="E31">A11</font></span></a></span>;<br/><a NAME="E7:47_1_1"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="ref" href="waybel_8.html#T5" target="_self" title="WAYBEL_8:th.5">Th5</a></span>;<br/><a NAME="E8:47_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E9:47_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">v9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/><a NAME="E10:47_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:47_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E17:47_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></span>;<br/><a NAME="E11:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A14</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_1"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/><a NAME="E12:47_1_1"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_1"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E13:47_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E5:47_1_1"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E14:47_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47_1_1"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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/></div><span class="kw">end;</span></div><a NAME="E19:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E20:47_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="waybel_8.html#D3" target="_self" title="WAYBEL_8:def.3">Def3</a></span>;<br/><a NAME="E21:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E14:47_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E16:47_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></span>;<br/><a NAME="E22:47_1"/>
<font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</a></span>;<br/><a NAME="E23:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <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 <font color="Maroon" title="c3">Imc</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><a NAME="E24:47_1"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> is   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="c3">Imc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:47_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">Imc</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">D</font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">Imc</font> ;<br/>
<a NAME="E3:47_1_2"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c3">Imc</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Imc</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> implies  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Imc</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47_1_2_1"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">and </span><br/><a NAME="E2:47_1_2_1"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Imc</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><a NAME="E3:47_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:47_1_2_1"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">and </span><br/><a NAME="E6:47_1_2_1"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:47_1_2_1"/><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><a NAME="E8:47_1_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_2_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:47_1_2_1"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">and </span><br/><a NAME="E11:47_1_2_1"/><span class="lab"><font color="Green" title="E42">A27</font></span>: 
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:47_1_2_1"/><span class="lab"><font color="Green" title="E43">A28</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">e</font> = <font color="Maroon" title="c11">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_2_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a></span>;<br/><a NAME="E14:47_1_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_2_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E15:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/><a NAME="E16:47_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A29</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E12:47_1_2_1"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">d</font> = <font color="Maroon" title="c10">d</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_2_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a></span>;<br/><a NAME="E18:47_1_2_1"/><span class="lab"><font color="Green" title="E45">A30</font></span>: 
<font color="Maroon" title="c13">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E19:47_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E20:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/><a NAME="E21:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E7:47_1_2_1"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></span>;<br/><a NAME="E22:47_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">e</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:47_1_2_1"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E23:47_1_2_1"/>
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/><a NAME="E24:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E25:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Imc</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c15">z</font> = <font color="Maroon" title="c14">z</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="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )</span><br/><a NAME="E27:47_1_2_1"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D14" title="WAYBEL_1:def.14">WAYBEL_1:def 14</a></span>;<br/><a NAME="E28:47_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T9" title="YELLOW_2:th.9">YELLOW_2:9</a></span>;<br/><a NAME="E29:47_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A33</font></span>: 
<font color="Maroon" title="c12">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>;<br/><a NAME="E30:47_1_2_1"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:47_1_2_1"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="ref" href="yellow_2.html#T9" title="YELLOW_2:th.9">YELLOW_2:9</a></span>;<br/><a NAME="E31:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>;<br/><a NAME="E32:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:47_1_2_1"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E33:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:47_1_2_1"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><a NAME="E34:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:47_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E17:47_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></span>;<br/><a NAME="E35:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></span>;<br/><a NAME="E36:47_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A34</font></span>: 
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/><a NAME="E37:47_1_2_1"/><span class="lab"><font color="Green" title="E50">A35</font></span>: 
<font color="Maroon" title="c12">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E38:47_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A36</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></span>;<br/><a NAME="E39:47_1_2_1"/>
<font color="Maroon" title="c12">e</font> is  <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47_1_2_1"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></span>;<br/><a NAME="E40:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">e</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></span>;<br/><a NAME="E41:47_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A37</font></span>: 
<font color="Maroon" title="c12">e</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:47_1_2_1"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></span>;<br/><a NAME="E42:47_1_2_1"/>
<font color="Maroon" title="c13">d</font> is  <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1_2_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></span>;<br/><a NAME="E43:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></span>;<br/><a NAME="E44:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:47_1_2_1"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></span>;<br/><a NAME="E45:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:47_1_2_1"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="waybel_3.html#T3" title="WAYBEL_3:th.3">WAYBEL_3:3</a></span>;<br/><a NAME="E46:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font> is  <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></span>;<br/><a NAME="E47:47_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A38</font></span>: 
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></span>;<br/><a NAME="E48:47_1_2_1"/>
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/><a NAME="E49:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E50:47_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:47_1_2_1"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E51:47_1_2_1"/>
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:47_1_2_1"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <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="c8">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )</span><br/><a NAME="E52:47_1_2_1"/>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E18:47_1_2_1"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E53:47_1_2_1"/>
( <font color="Maroon" title="c8">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_2_1"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E12:47_1_2_1"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E38:47_1_2_1"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</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/></div><span class="kw">end;</span></div>
<a NAME="E5:47_1_2"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T15" title="WAYBEL_3:th.15">WAYBEL_3:15</a></span>;<br/>
<a NAME="E6:47_1_2"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">c</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 <font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A39</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E8:47_1_2"/><span class="lab"><font color="Green" title="E37">A40</font></span>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">L</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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 <font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a>, <a class="ref" href="yellow_0.html#T42" title="YELLOW_0:th.42">YELLOW_0:42</a></span>;<br/>
<a NAME="E9:47_1_2"/><span class="lab"><font color="Green" title="E38">A41</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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 <font color="Maroon" title="c3">Imc</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> = 
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">L</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D11" title="YELLOW_0:def.11">YELLOW_0:def 11</a></span>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c3">Imc</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_1_2"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="txt" href="#E9:47_1_2"><span class="lab"><font color="Green" title="E38">A41</font></span></a>, <a class="ref" href="waybel_1.html#T55" title="WAYBEL_1:th.55">WAYBEL_1:55</a></span>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c3">Imc</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D11" title="YELLOW_0:def.11">YELLOW_0:def 11</a></span>
;<br/>
<a NAME="E11:47_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:47_1_2"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> is   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="c3">Imc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_2"><span class="lab"><font color="Green" title="E36">A39</font></span></a>, <a class="txt" href="#E4:47_1_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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/>


</div><span class="kw">end;</span></div><a NAME="E25:47_1"/><span class="lab"><font color="Green" title="E34">A42</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:47_1_3"/><span class="lab"><font color="Green" title="E35">A43</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 ; <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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">z1</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E3:47_1_3"/><span class="lab"><font color="Green" title="E36">A44</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_3"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">z2</font> = <font color="Maroon" title="c8">z1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Imc</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:47_1"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><a NAME="E5:47_1_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_3"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:47_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/><a NAME="E7:47_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">z2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></span>;<br/><a NAME="E8:47_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/><span class="kw">hence </span><a NAME="E9:47_1_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_3"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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/></div><span class="kw">end;</span></div><a NAME="E27:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A45</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E28:47_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:47_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></span>;<br/><a NAME="E29:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A46</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">Imc</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E14:47_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E23:47_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E21:47_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="waybel_1.html#T55" title="WAYBEL_1:th.55">WAYBEL_1:55</a></span>;<br/><a NAME="E30:47_1"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c6">a1</font></span>)</span>,<font color="Maroon" title="c3">Imc</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">Imc</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/><a NAME="E31:47_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p4">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">Imc</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:47_1"><span class="lab"><font color="Green" title="E36">A46</font></span></a>, <a class="txt" href="#E19:47_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E27:47_1"><span class="lab"><font color="Green" title="E35">A45</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Imc</font><span class="kw"> such that </span><br/><a NAME="E33:47_1"/><span class="lab"><font color="Green" title="E37">A47</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E34:47_1"/><span class="lab"><font color="Green" title="E38">A48</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E24:47_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="waybel_3.html#D1" title="WAYBEL_3:def.1">WAYBEL_3:def 1</a></span>;<br/><a NAME="E35:47_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:47_1"><span class="lab"><font color="Green" title="E37">A47</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E36:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A49</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/><a NAME="E37:47_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:47_1"><span class="lab"><font color="Green" title="E37">A47</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E38:47_1"/>
<font color="Maroon" title="c4">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:47_1"><span class="lab"><font color="Green" title="E38">A48</font></span></a>, <a class="txt" href="#E36:47_1"><span class="lab"><font color="Green" title="E39">A49</font></span></a>, <a class="ref" href="orders_2.html#T2" title="ORDERS_2:th.2">ORDERS_2:2</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/></div><span class="kw">end;</span></div>
<a NAME="E3:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A50</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E4:47"/>
<font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="waybel_8.html#T23" target="_self" title="WAYBEL_8:th.23">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:47"/>
<font color="Maroon" title="c2">c</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E22">A50</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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