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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="yellow_0.html#V1">lower-bounded</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V3">arithmetic</a> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel15.html#E4:13"><i><font color="Green">E14</font></i></a>, <a class="ref" href="waybel_0.html#T67">WAYBEL_0:67</a></i>;<br/>
<a NAME="E6:13"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E4:13"><i><font color="Green">E14</font></i></a>, <a class="ref" href="waybel_0.html#T68">WAYBEL_0:68</a></i>;<br/>
<a NAME="E7:13"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V2">algebraic</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_8.html#D5">WAYBEL_8:def 5</a></i>;<br/>
<a NAME="E8:13"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_8.html#V2">algebraic</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E1:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel15.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel13.html#T32">WAYBEL13:32</a></i>;<br/>
<a NAME="E9:13"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is  <a href="waybel_8.html#V3">arithmetic</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E2:13"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E10:13"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E1:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel13.html#T30">WAYBEL13:30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:13_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:13_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:13_1"/><i><font color="Green">E21</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E4:13_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_3.html#V1">compact</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_3.html#V1">compact</a> )
 <b>by </b><i><a class="txt" href="waybel15.html#E1:13_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel15.html#E2:13_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/><a NAME="E5:13_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_3.html#V1">compact</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_3.html#V1">compact</a> )
 <b>by </b><i><a class="txt" href="waybel15.html#E6:13"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel15.html#E9:13"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel15.html#E10:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="waybel13.html#T28">WAYBEL13:28</a></i>;<br/><a NAME="E6:13_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/><a NAME="E7:13_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><a NAME="E8:13_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:13_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a>
 <b>by </b><i><a class="txt" href="waybel15.html#E6:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/><a NAME="E10:13_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/><b>then </b><i><font color="Green">E25</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span> = 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="waybel15.html#E3:13_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i><a class="txt" href="waybel15.html#E8:13_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>
;<br/><a NAME="E12:13_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel15.html#E3:13_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="waybel15.html#E10:13_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/><a NAME="E13:13_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel15.html#E8:13_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/><a NAME="E14:13_1"/>
 <a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel15.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_8.html#D5">WAYBEL_8:def 5</a></i>;<br/><a NAME="E15:13_1"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel15.html#E6:13_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel15.html#E13:13_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="yellow_0.html#D16">YELLOW_0:def 16</a></i>;<br/><a NAME="E16:13_1"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/><a NAME="E17:13_1"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i><a class="txt" href="waybel15.html#E6:13"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel15.html#E9:13"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel15.html#E10:13"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel15.html#E11:13_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="waybel13.html#T28">WAYBEL13:28</a></i>;<br/><b>hence </b><a NAME="E18:13_1"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:13"/><b>then </b>
 <a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>2</sub></font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D16">YELLOW_0:def 16</a></i>;<br/>
<b>hence </b><a NAME="E13:13"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_8.html#V3">arithmetic</a>
 <b>by </b><i><a class="txt" href="waybel15.html#E8:13"><i><font color="Green">E16</font></i></a>, <a class="ref" href="waybel_8.html#D5">WAYBEL_8:def 5</a></i>;<br/>


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