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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>   <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">v</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">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp;  ex <font color="Olive" title="b4">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Olive" title="b2">F</font> st <font color="Olive" title="b4">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  holds <br/> ex <font color="Olive" title="b4">O</font> being  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b4">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">O</font> &amp; <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>   <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">v</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="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp;  ex <font color="Olive" title="b3">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">F</font> st <font color="Olive" title="b3">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  holds <br/> ex <font color="Olive" title="b3">O</font> being  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">O</font> &amp; <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</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="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp;  ex <font color="Olive" title="b2">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">F</font> st <font color="Olive" title="b2">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  holds <br/> ex <font color="Olive" title="b2">O</font> being  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp; <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp;  ex <font color="Olive" title="b1">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">F</font> st <font color="Olive" title="b1">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  implies  ex <font color="Olive" title="b1">O</font> being  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp; <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">O</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( for <font color="Olive" title="b1">A</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">F</font>  holds not <font color="Olive" title="b1">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  or  ex <font color="Olive" title="b1">O</font> being  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp; <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">O</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">V1</font> = <font color="Maroon" title="c2">V</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>   <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">G</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> </span>}</span>  as   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel12.html#T27" target="_self" title="WAYBEL12:th.27">Th27</a>, <a class="ref" href="waybel12.html#T28" target="_self" title="WAYBEL12:th.28">Th28</a>, <a class="ref" href="waybel12.html#T29" target="_self" title="WAYBEL12:th.29">Th29</a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c8">A</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">F</font><span class="kw"> such that </span><a NAME="E7:48"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c8">A</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">O</font> being  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp; <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">O</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c8">A</font><span class="kw"> such that </span><br/><a NAME="E9:48"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="card_3.html#T146" title="CARD_3:th.146">CARD_3:146</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 </span>}</span> ,<font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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"> such that </span><br/><a NAME="E12:48"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <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> ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> ex <font color="Olive" title="b3">z1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">z1</font> );<br/>
<a NAME="E13:48"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">AA</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">g</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></span>;<br/>
<a NAME="E15:48"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c12">AA</font> is    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E12:48"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="waybel12.html#T36" target="_self" title="WAYBEL12:th.36">Th36</a></span>;<br/>
<a NAME="E16:48"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> or <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">s</font> = <font color="Maroon" title="c13">q</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="E3:48_1"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_4.html#T42" title="YELLOW_4:th.42">YELLOW_4:42</a></span>;<br/>
<a NAME="E4:48_1"/>
<font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:48_1_1"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c16">t</font>
 <span class="kw">and </span><br/><a NAME="E4:48_1_1"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1"><span class="lab"><font color="Green" title="E38">A10</font></span></a></span>;<br/>
<a NAME="E5:48_1_1"/>
<font color="Maroon" title="c16">t</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E40">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:48_1_1"/>
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:48_1_1"><span class="lab"><font color="Green" title="E39">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:48_1"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:48"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>  ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c14">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<a NAME="E1:48_2"/>
<font color="Maroon" title="c12">AA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="waybel12.html#E39"><span class="lab"><font color="Green" title="E23">Lm4</font></span></a></span>;<br/>
<a NAME="E2:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c12">AA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E3:48_2"/>
<font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E4:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E38">A14</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">g1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E6:48_2"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g1</font>
 <span class="kw">and </span><br/><a NAME="E7:48_2"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 ;<br/>
<a NAME="E8:48_2"/>
<font color="Maroon" title="c15">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E9:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c15">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
;<br/>
<a NAME="E10:48_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b1">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c14">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c15">g1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_2"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="waybel_6.html#D1" title="WAYBEL_6:def.1">WAYBEL_6:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:48_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c14">x</font>,<font color="Olive" title="b1">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_2"><span class="lab"><font color="Green" title="E39">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c13">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c5">V1</font><span class="kw"> such that </span><br/><a NAME="E19:48"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font>
 <span class="kw">and </span><br/><a NAME="E20:48"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="#E17:48"><span class="lab"><font color="Green" title="E37">A13</font></span></a>);<br/></span>
<a NAME="E21:48"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E22:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A20</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">R</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">h</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">O</font> =  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>;<br/>
<a NAME="E24:48"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
<font color="Maroon" title="c14">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T62" title="WAYBEL_0:th.62">WAYBEL_0:62</a></span>;<br/>
<a NAME="E25:48"/><span class="lab"><font color="Green" title="E43">A22</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">y</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> holds <br/><font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c18">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> implies <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:48_3"/><span class="lab"><font color="Green" title="E44">A23</font></span>: 
( <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c20">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>, <font color="Maroon" title="c21">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:48_3"/><span class="lab"><font color="Green" title="E45">A24</font></span>: 
( <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">and </span><br/><a NAME="E4:48_3"/>
<font color="Maroon" title="c21">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:48_3"/><span class="lab"><font color="Green" title="E46">A25</font></span>: 
<font color="Maroon" title="c20">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c19">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">z1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48"><span class="lab"><font color="Green" title="E39">A18</font></span></a></span>;<br/>
<a NAME="E6:48_3"/><span class="lab"><font color="Green" title="E47">A26</font></span>: 
<font color="Maroon" title="c19">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/>
<a NAME="E7:48_3"/>
<font color="Maroon" title="c20">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_3"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="waybel_3.html#T1" title="WAYBEL_3:th.1">WAYBEL_3:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:48_3"/>
<font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_3"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E3:48_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E6:48_3"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:48"/><span class="lab"><font color="Green" title="E44">A27</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</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">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/><font color="Olive" title="b2">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">t</font>;<br/>
<a NAME="E1:48_4"/><span class="lab"><font color="Green" title="E45">A28</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_4_1"/><span class="lab"><font color="Green" title="E46">A29</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</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">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">k</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> holds <br/><font color="Olive" title="b2">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:48_4_1"/>
<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">s</font> = <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">k</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">let </span><font color="Maroon" title="c18">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</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="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c18">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Olive" title="b1">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c19">c</font>, <font color="Maroon" title="c20">d</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a</font> &amp; <font color="Maroon" title="c19">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c18">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c19">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">d</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:48_4_1"/><span class="lab"><font color="Green" title="E47">A30</font></span>: 
<font color="Maroon" title="c20">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a</font>
 <span class="kw">and </span><br/><a NAME="E5:48_4_1"/><span class="lab"><font color="Green" title="E48">A31</font></span>: 
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:48_4_1"/><span class="lab"><font color="Green" title="E49">A32</font></span>: 
<font color="Maroon" title="c18">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">d</font></span><br/>

<a NAME="E7:48_4_1"/><span class="lab"><font color="Green" title="E50">A33</font></span>: 
<font color="Maroon" title="c19">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:48"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E5:48_4_1"><span class="lab"><font color="Green" title="E48">A31</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_4_1_1"/>
( <font color="Maroon" title="c18">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> or <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_4_1"><span class="lab"><font color="Green" title="E49">A32</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_4_1_1_1"/>
<font color="Maroon" title="c18">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">d</font></span><br/><div class="add"><a NAME="E2:48_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_4_1"><span class="lab"><font color="Green" title="E46">A29</font></span></a>, <a class="txt" href="#E4:48_4_1"><span class="lab"><font color="Green" title="E47">A30</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_4_1_1_1"/>
<font color="Maroon" title="c19">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_4_1"><span class="lab"><font color="Green" title="E50">A33</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_4_1_1_2"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">d</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_4_1_1_2"/>
<font color="Maroon" title="c19">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_4_1"><span class="lab"><font color="Green" title="E47">A30</font></span></a>, <a class="txt" href="#E5:48_4_1"><span class="lab"><font color="Green" title="E48">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:48_4"/><span class="lab"><font color="Green" title="E46">A34</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/>
<a NAME="E3:48_4"/><span class="lab"><font color="Green" title="E47">A35</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:48_4"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="txt" href="#E1:48_4"><span class="lab"><font color="Green" title="E45">A28</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">y</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c19">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font> implies <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )</span><br/>



<span class="kw">assume </span><a NAME="E4:48_4"/>
( <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font></span><br/>

<span class="kw">hence </span><a NAME="E5:48_4"/>
<font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_4"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:48"/><span class="lab"><font color="Green" title="E45">A36</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="c14">R</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; not <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">y</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; not <font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">y</font> implies <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_5"/><span class="lab"><font color="Green" title="E46">A37</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 <span class="kw">and </span><br/><a NAME="E2:48_5"/><span class="lab"><font color="Green" title="E47">A38</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">y</font> or <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c18">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:48_5"/><span class="lab"><font color="Green" title="E48">A39</font></span>: 
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">and </span><br/><a NAME="E5:48_5"/><span class="lab"><font color="Green" title="E49">A40</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_5"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">m</font> = <font color="Maroon" title="c18">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_5"><span class="lab"><font color="Green" title="E48">A39</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:48_5"/><span class="lab"><font color="Green" title="E50">A41</font></span>: 
<font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">and </span><br/><a NAME="E9:48_5"/><span class="lab"><font color="Green" title="E51">A42</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_5"><span class="lab"><font color="Green" title="E46">A37</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">n</font> = <font color="Maroon" title="c20">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_5"><span class="lab"><font color="Green" title="E50">A41</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_5_1"/>
( <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">n</font> or <font color="Maroon" title="c21">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_5_1_1"/>
<font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">y</font> or <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_5_1_1"/>
( <font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">y</font> or <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:48"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="txt" href="#E9:48_5"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="txt" href="#E5:48_5"><span class="lab"><font color="Green" title="E49">A40</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_5_1_2"/>
<font color="Maroon" title="c21">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">y</font> or <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_5_1_2"/>
( <font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">y</font> or <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:48"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="txt" href="#E9:48_5"><span class="lab"><font color="Green" title="E51">A42</font></span></a>, <a class="txt" href="#E5:48_5"><span class="lab"><font color="Green" title="E49">A40</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E28:48"/><span class="lab"><font color="Green" title="E46">A43</font></span>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> is  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_6.html#D1" title="WAYBEL_6:def.1">WAYBEL_6:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_6"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:48_6"/><span class="lab"><font color="Green" title="E47">A44</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">and </span><br/><a NAME="E4:48_6"/><span class="lab"><font color="Green" title="E48">A45</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">Y</font> being   <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 <font color="Maroon" title="c14">R</font><span class="kw"> such that </span><br/><a NAME="E6:48_6"/><span class="lab"><font color="Green" title="E49">A46</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c18">Y</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E7:48_6"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c18">Y</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_6"><span class="lab"><font color="Green" title="E48">A45</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_6_1"/>
( <font color="Maroon" title="c18">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c18">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_6_1_1"/>
<font color="Maroon" title="c18">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font> )</span><br/><div class="add"><a NAME="E2:48_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:48"><span class="lab"><font color="Green" title="E45">A36</font></span></a>, <a class="txt" href="#E6:48_6"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="ref" href="waybel12.html#T31" target="_self" title="WAYBEL12:th.31">Th31</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:48_6_1_1"/><span class="lab"><font color="Green" title="E50">A47</font></span>: 
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">and </span><br/><a NAME="E5:48_6_1_1"/><span class="lab"><font color="Green" title="E51">A48</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_6_1_1"><span class="lab"><font color="Green" title="E50">A47</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">x1</font>, <font color="Maroon" title="c22">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>, <font color="Maroon" title="c23">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E8:48_6_1_1"/><span class="lab"><font color="Green" title="E52">A49</font></span>: 
<font color="Maroon" title="c21">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">and </span><br/><a NAME="E9:48_6_1_1"/><span class="lab"><font color="Green" title="E53">A50</font></span>: 
<font color="Maroon" title="c22">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:48_6_1_1"/>
<font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:48_6_1_1"/><span class="lab"><font color="Green" title="E54">A51</font></span>: 
<font color="Maroon" title="c22">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c21">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c23">z1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48"><span class="lab"><font color="Green" title="E39">A18</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c22">y1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> &amp; <font color="Maroon" title="c22">y1</font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font> )</span><br/><a NAME="E12:48_6_1_1"/>
<font color="Maroon" title="c22">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E9:48_6_1_1"><span class="lab"><font color="Green" title="E53">A50</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E13:48_6_1_1"/>
<font color="Maroon" title="c22">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48"><span class="lab"><font color="Green" title="E42">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">y1</font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font></span><br/><a NAME="E14:48_6_1_1"/>
<font color="Maroon" title="c21">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c23">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/><a NAME="E15:48_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c21">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_6_1_1"><span class="lab"><font color="Green" title="E54">A51</font></span></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></span>;<br/><span class="kw">hence </span><a NAME="E16:48_6_1_1"/>
<font color="Maroon" title="c22">y1</font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_6"><span class="lab"><font color="Green" title="E47">A44</font></span></a>, <a class="txt" href="#E5:48_6_1_1"><span class="lab"><font color="Green" title="E51">A48</font></span></a>, <a class="txt" href="#E8:48_6_1_1"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_6_1_2"/><span class="lab"><font color="Green" title="E50">A52</font></span>: 
<font color="Maroon" title="c18">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c19">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:48_6_1_2"/><span class="lab"><font color="Green" title="E51">A53</font></span>: 
<font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">b</font> = <font color="Maroon" title="c19">b</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="#E3:48_6_1_2"><span class="lab"><font color="Green" title="E51">A53</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:48_6_1_2"/><span class="lab"><font color="Green" title="E52">A54</font></span>: 
<font color="Maroon" title="c21">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">and </span><br/><a NAME="E7:48_6_1_2"/>
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_6_1_2"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">n</font> = <font color="Maroon" title="c21">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_6_1_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a></span>;<br/><a NAME="E9:48_6_1_2"/><span class="lab"><font color="Green" title="E53">A55</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">x1</font>, <font color="Maroon" title="c24">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>, <font color="Maroon" title="c25">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E11:48_6_1_2"/>
<font color="Maroon" title="c23">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">and </span><br/><a NAME="E12:48_6_1_2"/><span class="lab"><font color="Green" title="E54">A56</font></span>: 
<font color="Maroon" title="c24">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:48_6_1_2"/>
<font color="Maroon" title="c25">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:48_6_1_2"/><span class="lab"><font color="Green" title="E55">A57</font></span>: 
<font color="Maroon" title="c24">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c23">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c25">z1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48"><span class="lab"><font color="Green" title="E39">A18</font></span></a></span>;<br/><a NAME="E15:48_6_1_2"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_6"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="txt" href="#E1:48_6_1_2"><span class="lab"><font color="Green" title="E50">A52</font></span></a>, <a class="ref" href="yellow_0.html#D12" title="YELLOW_0:def.12">YELLOW_0:def 12</a></span>;<br/><a NAME="E16:48_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_6"><span class="lab"><font color="Green" title="E47">A44</font></span></a>, <a class="txt" href="#E9:48_6_1_2"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></span>;<br/><a NAME="E17:48_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A58</font></span>: 
<font color="Maroon" title="c23">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c24">y1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> &amp; <font color="Maroon" title="c24">y1</font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font> )</span><br/><a NAME="E18:48_6_1_2"/>
<font color="Maroon" title="c24">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E12:48_6_1_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E19:48_6_1_2"/>
<font color="Maroon" title="c24">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48"><span class="lab"><font color="Green" title="E42">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">y1</font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font></span><br/><a NAME="E20:48_6_1_2"/>
<font color="Maroon" title="c23">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c25">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c23">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/><a NAME="E21:48_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c23">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_6_1_2"><span class="lab"><font color="Green" title="E55">A57</font></span></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></span>;<br/><span class="kw">hence </span><a NAME="E22:48_6_1_2"/>
<font color="Maroon" title="c24">y1</font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:48_6_1_2"><span class="lab"><font color="Green" title="E56">A58</font></span></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E29:48"/><span class="lab"><font color="Green" title="E47">A59</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</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="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c17">a</font>, <font color="Maroon" title="c18">b</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon" title="c18">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">a</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:48_7"/><span class="lab"><font color="Green" title="E48">A60</font></span>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">a</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c19">gn</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> </span>}</span> , <font color="Maroon" title="c20">gn1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="waybel12.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E3:48_7"/><span class="lab"><font color="Green" title="E49">A61</font></span>: 
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c19">gn</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c20">gn1</font>,<font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T55" title="YELLOW_0:th.55">YELLOW_0:55</a></span>;<br/>
<a NAME="E4:48_7"/><span class="lab"><font color="Green" title="E50">A62</font></span>: 
<font color="Maroon" title="c19">gn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">gn1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">gn</font> or <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">gn1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_7_1"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">gn</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">gn1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:48_7_1"/><span class="lab"><font color="Green" title="E51">A63</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">k</font>
 <span class="kw">and </span><br/><a NAME="E4:48_7_1"/><span class="lab"><font color="Green" title="E52">A64</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 ;<br/>
<a NAME="E5:48_7_1"/>
<font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_7_1"><span class="lab"><font color="Green" title="E52">A64</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:48_7_1"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">gn1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_7_1"><span class="lab"><font color="Green" title="E51">A63</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:48_7"/>
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c19">gn</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c20">gn1</font>,<font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E1:48_7"><span class="lab"><font color="Green" title="E48">A60</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:48_7"/>
<font color="Maroon" title="c18">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_7"><span class="lab"><font color="Green" title="E49">A61</font></span></a>, <a class="txt" href="#E4:48_7"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="ref" href="yellow_0.html#T35" title="YELLOW_0:th.35">YELLOW_0:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:48"/><span class="lab"><font color="Green" title="E48">A65</font></span>: 
<font color="Maroon" title="c12">AA</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c14">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_4.html#D2" title="YELLOW_4:def.2">YELLOW_4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">a</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_8"/>
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">a</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:48_8"/><span class="lab"><font color="Green" title="E49">A66</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">and </span><br/><a NAME="E4:48_8"/><span class="lab"><font color="Green" title="E50">A67</font></span>: 
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_8"><span class="lab"><font color="Green" title="E49">A66</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c20">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>, <font color="Maroon" title="c21">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:48_8"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">and </span><br/><a NAME="E8:48_8"/><span class="lab"><font color="Green" title="E51">A68</font></span>: 
<font color="Maroon" title="c20">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:48_8"/><span class="lab"><font color="Green" title="E52">A69</font></span>: 
<font color="Maroon" title="c21">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:48_8"/><span class="lab"><font color="Green" title="E53">A70</font></span>: 
<font color="Maroon" title="c20">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c19">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">z1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48"><span class="lab"><font color="Green" title="E39">A18</font></span></a></span>;<br/>
<a NAME="E11:48_8"/>
( <font color="Maroon" title="c19">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">z1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_8"><span class="lab"><font color="Green" title="E53">A70</font></span></a>, <a class="ref" href="waybel_3.html#T1" title="WAYBEL_3:th.1">WAYBEL_3:1</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/>
<a NAME="E12:48_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A71</font></span>: 
<font color="Maroon" title="c20">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/>
<a NAME="E13:48_8"/><span class="lab"><font color="Green" title="E55">A72</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E14:48_8"/>
<font color="Maroon" title="c21">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:48"><span class="lab"><font color="Green" title="E47">A59</font></span></a>, <a class="txt" href="#E4:48_8"><span class="lab"><font color="Green" title="E50">A67</font></span></a>, <a class="txt" href="#E9:48_8"><span class="lab"><font color="Green" title="E52">A69</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:48_8"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; <font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_8"><span class="lab"><font color="Green" title="E51">A68</font></span></a>, <a class="txt" href="#E13:48_8"><span class="lab"><font color="Green" title="E55">A72</font></span></a>, <a class="txt" href="#E12:48_8"><span class="lab"><font color="Green" title="E54">A71</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c16">O</font> =  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> as   <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:48"><span class="lab"><font color="Green" title="E46">A43</font></span></a></span>;<br/>
<a NAME="E32:48"/>
<font color="Maroon" title="c14">R</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c16">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:48"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="waybel12.html#T20" target="_self" title="WAYBEL12:th.20">Th20</a></span>;<br/>
<a NAME="E33:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A73</font></span>: 
<font color="Maroon" title="c12">AA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:48"><span class="lab"><font color="Green" title="E48">A65</font></span></a>, <a class="ref" href="yellow_4.html#T7" title="YELLOW_4:th.7">YELLOW_4:7</a>, <a class="ref" href="yellow_4.html#T8" title="YELLOW_4:th.8">YELLOW_4:8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">O</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">O</font> &amp; <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">O</font> )</span><br/>

<span class="kw">thus </span><a NAME="E34:48"/>
<font color="Maroon" title="c16">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">O</font> &amp; <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">O</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">O</font> or <font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_9"/>
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">O</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:48_9"/><span class="lab"><font color="Green" title="E50">A74</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">q</font>
 <span class="kw">and </span><br/><a NAME="E5:48_9"/><span class="lab"><font color="Green" title="E51">A75</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">Y</font> being   <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 <font color="Maroon" title="c14">R</font><span class="kw"> such that </span><br/><a NAME="E7:48_9"/><span class="lab"><font color="Green" title="E52">A76</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c20">Y</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E8:48_9"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c20">Y</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_9"><span class="lab"><font color="Green" title="E51">A75</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_9_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_9_1"/>
( <font color="Maroon" title="c20">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c20">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_9_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_9_1_1"/>
<font color="Maroon" title="c20">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font></span><br/><div class="add"><a NAME="E2:48_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:48"><span class="lab"><font color="Green" title="E45">A36</font></span></a>, <a class="txt" href="#E7:48_9"><span class="lab"><font color="Green" title="E52">A76</font></span></a>, <a class="ref" href="waybel12.html#T31" target="_self" title="WAYBEL12:th.31">Th31</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:48_9_1_1"/><span class="lab"><font color="Green" title="E53">A77</font></span>: 
<font color="Maroon" title="c21">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">and </span><br/><a NAME="E5:48_9_1_1"/><span class="lab"><font color="Green" title="E54">A78</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">n</font> = <font color="Maroon" title="c21">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_9_1_1"><span class="lab"><font color="Green" title="E53">A77</font></span></a></span>;<br/><a NAME="E7:48_9_1_1"/>
 ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> ex <font color="Olive" title="b3">z1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">z1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48"><span class="lab"><font color="Green" title="E39">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:48_9_1_1"/>
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_9"><span class="lab"><font color="Green" title="E50">A74</font></span></a>, <a class="txt" href="#E5:48_9_1_1"><span class="lab"><font color="Green" title="E54">A78</font></span></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_9_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_9_1_2"/><span class="lab"><font color="Green" title="E53">A79</font></span>: 
<font color="Maroon" title="c20">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font></span><br/><div class="add"><a NAME="E2:48_9_1_2"/><span class="lab"><font color="Green" title="E54">A80</font></span>: 
<font color="Maroon" title="c18">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</a></span>;<br/><a NAME="E3:48_9_1_2"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_9"><span class="lab"><font color="Green" title="E52">A76</font></span></a>, <a class="txt" href="#E1:48_9_1_2"><span class="lab"><font color="Green" title="E53">A79</font></span></a>, <a class="ref" href="yellow_0.html#D12" title="YELLOW_0:def.12">YELLOW_0:def 12</a></span>;<br/><a NAME="E4:48_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_9"><span class="lab"><font color="Green" title="E50">A74</font></span></a>, <a class="txt" href="#E2:48_9_1_2"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_9_1_2"/>
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel12.html#T12" target="_self" title="WAYBEL12:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E35:48"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:48"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E22:48"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E24:48"><span class="lab"><font color="Green" title="E42">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">O</font></span><br/>

<a NAME="E36:48"/>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">AA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="waybel12.html#D3" target="_self" title="WAYBEL12:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E37:48"/>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:48"><span class="lab"><font color="Green" title="E49">A73</font></span></a>, <a class="ref" href="waybel_0.html#T62" title="WAYBEL_0:th.62">WAYBEL_0:62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
