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

<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>  st <font color="Maroon" title="c1">S</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  holds <br/> ex <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>  &amp; ( for <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="c2">T</font> holds  <font color="Olive" title="b2">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <font color="Olive" title="b1">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b3">t</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">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" title="c1">S</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  &amp; <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  implies  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<font color="Olive" title="b1">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>  &amp; ( for <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="c2">T</font> holds  <font color="Olive" title="b1">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b2">t</font></span>)</span> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c1">S</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> 
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</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">d</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<font color="Olive" title="b1">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>  &amp; ( for <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="c2">T</font> holds  <font color="Olive" title="b1">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b2">t</font></span>)</span> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> );<br/>
<a NAME="E3:31"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> implies  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Olive" title="b1">u</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Olive" title="b1">u</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<span class="kw">take </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e</font>, <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span>] )</span><br/>

<span class="kw">thus </span><a NAME="E3:31_1"/>
(  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e</font>, <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</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="c4">d</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E5:31"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E25">A3</font></span></a>);<br/></span>
<a NAME="E6:31"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> holds  <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:31_2"/>
 ex <font color="Olive" title="b1">t1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">t1</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E26">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:31_2"/>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</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="c5">d</font> = <font color="Maroon" title="c4">d</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E8:31"/>
for <font color="Olive" title="b1">X</font> being  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Olive" title="b1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a></span>;<br/>
<a NAME="E9:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c3">g</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T69" title="WAYBEL_0:th.69">WAYBEL_0:69</a></span>;<br/>
<a NAME="E10:31"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> iff <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</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">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> iff <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</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="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> iff <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s</font> )</span><br/>



<a NAME="E1:31_3"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T39" title="WAYBEL_0:th.39">WAYBEL_0:39</a></span>;<br/>
<a NAME="E2:31_3"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>,<font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/>
<a NAME="E3:31_3"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></span>;<br/>
<a NAME="E4:31_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/>
<div><span class="kw">hereby </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="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s</font> implies <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:31_3_1"/>
<font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</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="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s</font></span><br/><a NAME="E2:31_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></span>;<br/><a NAME="E3:31_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></span>;<br/><span class="kw">hence </span><a NAME="E4:31_3_1"/>
<font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_3"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</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="E6:31_3"/>
<font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a></span>;<br/>
<a NAME="E7:31_3"/><span class="kw">then </span>
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_3"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></span>;<br/>
<a NAME="E8:31_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_3"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a>, <a class="ref" href="yellow_0.html#T35" title="YELLOW_0:th.35">YELLOW_0:35</a></span>;<br/>
<a NAME="E9:31_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T39" title="WAYBEL_0:th.39">WAYBEL_0:39</a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:31_3"/>
<font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s</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="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span><br/>

<a NAME="E11:31_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="waybel_1.html#D2" target="_self" title="WAYBEL_1:def.2">Def2</a></span>;<br/>
<a NAME="E12:31_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:31_3"/>
<font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31_3"><span class="lab"><font color="Green" title="E33">A11</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">take </span>
<font color="Maroon" title="c5">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>  &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> holds  <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">t</font></span>)</span> ) )</span><br/>

<a NAME="E11:31"/>
<font color="Maroon" title="c5">d</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">t1</font>, <font color="Maroon" title="c7">t2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_1.html#D2" target="_self" title="WAYBEL_1:def.2">WAYBEL_1:def 2</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="c6">t1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">t2</font> implies <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">t2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_4"/>
<font color="Maroon" title="c6">t1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">t2</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="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">t2</font></span><br/>

<a NAME="E2:31_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A12</font></span>: 
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c7">t2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T22" title="WAYBEL_0:th.22">WAYBEL_0:22</a></span>;<br/>
<a NAME="E3:31_4"/>
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t1</font></span>)</span>,<font color="Maroon" title="c1">S</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c7">t2</font></span>)</span>,<font color="Maroon" title="c1">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/>
<a NAME="E4:31_4"/><span class="kw">then </span>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t1</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c7">t2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_4"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a>, <a class="ref" href="yellow_0.html#T35" title="YELLOW_0:th.35">YELLOW_0:35</a></span>;<br/>
<a NAME="E5:31_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c7">t2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31_4"/>
<font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E27">A5</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="E12:31"/>
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="waybel_1.html#T9" target="_self" title="WAYBEL_1:th.9">Th9</a></span>; <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">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> holds  <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">t</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E13:31"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>,<font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_1.html#D6" target="_self" title="WAYBEL_1:def.6">WAYBEL_1:def 6</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="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">S</font> &amp;  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E14:31"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E27">A5</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">  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span><br/>

<a NAME="E15:31"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T39" title="WAYBEL_0:th.39">WAYBEL_0:39</a></span>;<br/>
<a NAME="E16:31"/>
<font color="Maroon" title="c3">g</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a></span>;<br/>
<a NAME="E17:31"/><span class="kw">then </span>
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:31"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></span>;<br/>
<a NAME="E18:31"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:31"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a>, <a class="ref" href="yellow_0.html#T35" title="YELLOW_0:th.35">YELLOW_0:35</a></span>;<br/>
<a NAME="E19:31"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T39" title="WAYBEL_0:th.39">WAYBEL_0:39</a></span>;<br/>
<a NAME="E20:31"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/>
<a NAME="E21:31"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:31"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:31"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</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>
