<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">L2</font> #) &amp; <font color="Maroon" title="c1">L1</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a> implies <font color="Maroon" title="c2">L2</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">L2</font> #)
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c1">L1</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L2</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a></span><br/>

<a NAME="E3:31"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c1">L1</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E2:31"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="waybel_8.html#D5" target="_self" title="WAYBEL_8:def.5">Def5</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green" title="E23">A4</font></i>: 
<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="waybel_8.html#E1:31"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="waybel_8.html#T17" target="_self" title="WAYBEL_8:th.17">Th17</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green" title="E24">A5</font></i>: 
 <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L1</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E2:31"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="waybel_8.html#D5" target="_self" title="WAYBEL_8:def.5">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">y2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x2</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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L2</font></span>)</span> &amp; <font color="Maroon" title="c4">y2</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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L2</font></span>)</span> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c2">L2</font> implies  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L2</font></span>)</span> )</span><br/><b>reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="waybel_8.html#E1:31"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:31_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c3">x2</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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L2</font></span>)</span>
 <b>and </b><br/><a NAME="E3:31_1"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c4">y2</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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L2</font></span>)</span>
 <b>and </b><br/><a NAME="E4:31_1"/><i><font color="Green" title="E27">A8</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c2">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L2</font></span>)</span></span><br/><a NAME="E5:31_1"/><i><font color="Green" title="E28">A9</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="waybel_8.html#E4:31"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self" title="WAYBEL_8:def.4">Def4</a></i>;<br/><a NAME="E6:31_1"/><i><font color="Green" title="E29">A10</font></i>: 
<font color="Maroon" title="c1">L1</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:31"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self" title="WAYBEL_8:def.4">Def4</a></i>;<br/><a NAME="E7:31_1"/>
( <font color="Maroon" title="c3">x2</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> &amp; <font color="Maroon" title="c4">y2</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:31_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="waybel_8.html#E3:31_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></i>;<br/><a NAME="E8:31_1"/><b>then </b>
( <font color="Maroon" title="c5">x1</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> &amp; <font color="Maroon" title="c6">y1</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:31"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="waybel_8.html#E5:31_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="waybel_8.html#T9" target="_self" title="WAYBEL_8:th.9">Th9</a></i>;<br/><a NAME="E9:31_1"/><b>then </b><i><font color="Green" title="E30">A11</font></i>: 
( <font color="Maroon" title="c5">x1</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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L1</font></span>)</span> &amp; <font color="Maroon" title="c6">y1</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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></i>;<br/><a NAME="E10:31_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L1</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:31"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="waybel_8.html#E4:31_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="yellow_0.html#T14" title="YELLOW_0:th.14">YELLOW_0:14</a></i>;<br/><a NAME="E11:31_1"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L1</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E5:31"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="waybel_8.html#E9:31_1"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="yellow_0.html#D16" title="YELLOW_0:def.16">YELLOW_0:def 16</a></i>;<br/><a NAME="E12:31_1"/><b>then </b><i><font color="Green" title="E31">A12</font></i>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> 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#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></i>;<br/><a NAME="E13:31_1"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:31"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="waybel_8.html#E4:31_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="yellow_0.html#T27" title="YELLOW_0:th.27">YELLOW_0:27</a></i>;<br/><a NAME="E14:31_1"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:31"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="waybel_8.html#E6:31_1"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="waybel_8.html#E12:31_1"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="waybel_8.html#T9" target="_self" title="WAYBEL_8:th.9">Th9</a></i>;<br/><b>hence </b><a NAME="E15:31_1"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <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 <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L2</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self" title="WAYBEL_8:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:31"/><b>then </b>
 <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">L2</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D16" title="YELLOW_0:def.16">YELLOW_0:def 16</a></i>;<br/>
<b>hence </b><a NAME="E8:31"/>
<font color="Maroon" title="c2">L2</font> is <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:31"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="waybel_8.html#D5" target="_self" title="WAYBEL_8:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
