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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">l</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b2">X</font> &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> holds <br/><font color="Olive" title="b3">x</font> is  <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>  ) ) ) implies  for <font color="Olive" title="b1">l</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">l</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"><span class="p2">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( $2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> ($2,<font color="Maroon" title="c1">L</font>) );<br/>
<span class="kw">assume </span><a NAME="E1:56"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b2">X</font> &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> holds <br/><font color="Olive" title="b3">x</font> is  <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</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">l</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">l</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"><span class="p2">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>)</span><br/>

<a NAME="E2:56"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> holds <br/> ex <font color="Olive" title="b2">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">e</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="c2">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> implies  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">e</font>,<font color="Olive" title="b1">u</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1"/>
<font color="Maroon" title="c2">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>
 ; <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">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">e</font>,<font color="Olive" title="b1">u</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">l</font> = <font color="Maroon" title="c2">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:56_1"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E5:56_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 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">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Olive" title="b1">x</font> is  <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E36">A1</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">p1</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="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:56_1_1"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c5">p1</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="E3:56_1_1"/>
<font color="Maroon" title="c6">p</font> is  <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E1:56_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a></span>;<br/><a NAME="E4:56_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  is  <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_6.html#D8" target="_self" title="WAYBEL_6:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E5:56_1_1"/>
<font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_6.html#D7" target="_self" title="WAYBEL_6:def.7">Def7</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="E7:56_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></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/>
<span class="kw">hence </span><a NAME="E8:56_1"/>
 ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">e</font>,<font color="Olive" title="b1">u</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1"><span class="lab"><font color="Green" title="E38">A3</font></span></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>
<span class="kw">consider </span><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:56"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</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>
 <span class="kw">and </span><br/><a NAME="E5:56"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E37">A2</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c3">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="c3">l</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"><span class="p2">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>)</span><br/>

<a NAME="E6:56"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</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="E7:56"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
 <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> ) </span> } </span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> or <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_2"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">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="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> ) </span> } </span> </span><br/>

<a NAME="E2:56_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E39">A7</font></span></a></span>;<br/>
<a NAME="E3:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E4:56_2"/>
( <font color="Maroon" title="c4">e</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">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font></span>)</span>,<font color="Maroon" title="c1">L</font>) &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:56_2"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E43">A11</font></span></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>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> );<br/>
<a NAME="E8:56"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c3">l</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_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#D5" title="WAYBEL_3:def.5">WAYBEL_3:def 5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">Z</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</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#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> ) </span> } </span> ;<br/>
<a NAME="E9:56"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</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#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</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#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> ) </span> } </span>  or <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_3"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</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#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</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="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:56_3"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <span class="kw">and </span><br/><a NAME="E4:56_3"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E6:56_3"/><span class="lab"><font color="Green" title="E46">A16</font></span>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">and </span><br/><a NAME="E7:56_3"/><span class="lab"><font color="Green" title="E47">A17</font></span>: 
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E8:56_3"/><span class="lab"><font color="Green" title="E48">A18</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_3"><span class="lab"><font color="Green" title="E45">A15</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">e1</font> = <font color="Maroon" title="c5">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="#E3:56_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E6:56_3"><span class="lab"><font color="Green" title="E46">A16</font></span></a></span>;<br/>
<a NAME="E10:56_3"/>
<font color="Maroon" title="c8">e1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E6:56_3"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="yellow_2.html#T22" title="YELLOW_2:th.22">YELLOW_2:22</a></span>;<br/>
<a NAME="E11:56_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A19</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_3"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></span>;<br/>
<a NAME="E12:56_3"/>
 ex <font color="Olive" title="b1">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_3"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E13:56_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E39">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:56_3"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E6:56_3"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="txt" href="#E11:56_3"><span class="lab"><font color="Green" title="E49">A19</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="E10:56"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</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#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> ) </span> } </span> ,<font color="Maroon" title="c1">L</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="E11:56"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c1">L</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="E12:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A21</font></span>: 
 <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_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</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"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span>,<font color="Maroon" title="c1">L</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></span>;<br/>
<a NAME="E13:56"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span>  or <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_4"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</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="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span><br/>

<a NAME="E2:56_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:56_4"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font>
 ; <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>
<span class="kw">then </span><font color="Maroon" title="c3">l</font> = 
 <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="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> ,<font color="Maroon" title="c1">L</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> </span>)</span>,<font color="Maroon" title="c1">L</font>)
<span class="kw">from</span> <span class="lab"><a class="ref" href="yellow_3.html#S5" title="YELLOW_3:sch.5">YELLOW_3:sch 5</a>()</span>
;<br/>
<a NAME="E15:56"/><span class="kw">then </span>
<font color="Maroon" title="c3">l</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_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E6:56"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E9:56"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:56"/>
<font color="Maroon" title="c3">l</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"><span class="p2">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="txt" href="#E12:56"><span class="lab"><font color="Green" title="E45">A21</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>
