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<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be   <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c1">L1</font> is <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> &amp; <font color="Maroon" title="c1">L1</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> )
 ;<br/>

<a NAME="E3:39"/><i><font color="Green" title="E36">A3</font></i>: 
( ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> holds <br/> ( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) ) &amp; <font color="Maroon" title="c1">L1</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">L1</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a> )
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E5:39"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="waybel13.html#E5:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="waybel_0.html#T67" title="WAYBEL_0:th.67">WAYBEL_0:67</a></i>;<br/>
<a NAME="E7:39"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E5:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="waybel_0.html#T68" title="WAYBEL_0:th.68">WAYBEL_0:68</a></i>;<br/>
<div><i><font color="Green" title="E39">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/><a NAME="E1:39_1"/><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E2:39_1"/>
( not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">y</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="c2">L2</font> )
 ;<br/><a NAME="E3:39_1"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:39_1"/><b>then </b><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:39_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E5:39_1"/><i><font color="Green" title="E42">A9</font></i>: 
( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 <b>by </b><i><a class="txt" href="waybel13.html#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/><a NAME="E6:39_1"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c1">L1</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/><a NAME="E7:39_1"/><b>then </b><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c2">L2</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="waybel13.html#T30" target="_self" title="WAYBEL13:th.30">Th30</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">Y</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">Y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1"/><i><font color="Green" title="E45">A12</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E4:39_1_1_1"/><i><font color="Green" title="E46">A13</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>and </b><br/><a NAME="E5:39_1_1_1"/><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">v</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">v</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="c2">L2</font> <b>by </b><i><a class="txt" href="waybel13.html#E3:39_1_1_1"><i><font color="Green" title="E45">A12</font></i></a></i>;<br/><a NAME="E7:39_1_1_1"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:39_1_1_1"><i><font color="Green" title="E46">A13</font></i></a></i>;<br/><a NAME="E8:39_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">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="c5">y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:39_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="waybel13.html#E4:39_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/><a NAME="E9:39_1_1_1"/><b>then </b><i><font color="Green" title="E48">A15</font></i>: 
( <font color="Maroon" title="c9">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c9">v</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 <b>by </b><i><a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/><a NAME="E10:39_1_1_1"/><b>then </b><i><font color="Green" title="E49">A16</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E7:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="waybel13.html#E6:39_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="waybel13.html#E7:39_1"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="waybel13.html#T28" target="_self" title="WAYBEL13:th.28">Th28</a></i>;<br/><a NAME="E11:39_1_1_1"/>
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E7:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="waybel13.html#E9:39_1_1_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E12:39_1_1_1"/>
<font color="Maroon" title="c7">z</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:39_1_1_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="txt" href="waybel13.html#E10:39_1_1_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c4">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">Y</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><b> such that </b><br/><a NAME="E4:39_1_1"/><i><font color="Green" title="E45">A17</font></i>: 
<font color="Maroon" title="c8">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <b>and </b><br/><a NAME="E5:39_1_1"/><i><font color="Green" title="E46">A18</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:39_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></i>;<br/><b>take </b><font color="Maroon" title="c9">fx</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>;<br/><a NAME="E6:39_1_1"/><i><font color="Green" title="E47">A19</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c8">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 <b>by </b><i><a class="txt" href="waybel13.html#E4:39_1_1"><i><font color="Green" title="E45">A17</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/><a NAME="E7:39_1_1"/><b>then </b><i><font color="Green" title="E48">A20</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="waybel13.html#E6:39_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="waybel13.html#E7:39_1"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="waybel13.html#T28" target="_self" title="WAYBEL13:th.28">Th28</a></i>;<br/><a NAME="E8:39_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="waybel13.html#E6:39_1_1"><i><font color="Green" title="E47">A19</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E9:39_1_1"/>
<font color="Maroon" title="c9">fx</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E7:39_1_1"><i><font color="Green" title="E48">A20</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/><a NAME="E10:39_1_1"/>
<font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:39_1_1"/><b>then </b><i><font color="Green" title="E49">A21</font></i>: 
<font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font> = 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">Y</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E1:39_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>
<br/>.= 
<font color="Maroon" title="c6">Y</font>
<b>by </b><i><a class="txt" href="waybel13.html#E11:39_1_1"><i><font color="Green" title="E49">A21</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>
;<br/><b>hence </b><a NAME="E13:39_1_1"/>
<font color="Maroon" title="c9">fx</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="waybel13.html#E5:39_1_1"><i><font color="Green" title="E46">A18</font></i></a>, <a class="ref" href="waybel13.html#T19" target="_self" title="WAYBEL13:th.19">Th19</a></i>;<br/></div><b>end;</b></div><a NAME="E9:39_1"/><b>then </b>
( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></i>;<br/><b>hence </b><a NAME="E10:39_1"/>
( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c5">y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c5">y</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 <b>by </b><i><a class="txt" href="waybel13.html#E1:39_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="waybel13.html#E4:39_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:39"/><i><font color="Green" title="E40">A22</font></i>: 
<font color="Maroon" title="c2">L2</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="waybel13.html#E3:39"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="waybel13.html#T30" target="_self" title="WAYBEL13:th.30">Th30</a></i>;<br/>
<a NAME="E10:39"/>
<font color="Maroon" title="c1">L1</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/>
<a NAME="E11:39"/><b>then </b>
<font color="Maroon" title="c2">L2</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="waybel13.html#E3:39"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="waybel13.html#T31" target="_self" title="WAYBEL13:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E12:39"/>
<font color="Maroon" title="c2">L2</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E8:39"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="waybel13.html#E9:39"><i><font color="Green" title="E40">A22</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/>


</div>
