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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being  <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a><br/> for <font color="Olive" title="b2">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Olive" title="b1">Y</font></span>)</span>  st ( for <font color="Olive" title="b3">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Olive" title="b2">N</font>,<font color="Olive" title="b3">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Olive" title="b1">Y</font></span>)</span> ) holds <br/> <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b2">N</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Olive" title="b1">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c2">Y</font> be   <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>  st ( for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Olive" title="b1">N</font>,<font color="Olive" title="b2">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> ) holds <br/> <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b1">N</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c3">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> ) implies  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> )</span><br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E30">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><br/>

<b>set </b><font color="Maroon" title="c4">n</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>;<br/>
<b>set </b><font color="Maroon" title="c5">L</font> = <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <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 <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> =  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,$1,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c6">f</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="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span><b> such that </b><br/><a NAME="E3:53"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">u</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19" title="WAYBEL24:th.19">WAYBEL24:19</a></i>;<br/>
<a NAME="E5:53"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> st <br/>(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<a NAME="E1:53_1"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:53_1"/><i><font color="Green" title="E33">A4</font></i>: 
<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">a</font> &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font> ) )</span><br/>

<b>thus </b><a NAME="E3:53_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> or <font color="Maroon" title="c8">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">a</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>
 <b>and </b><br/><a NAME="E4:53_1_1"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">i</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="waybel25.html#E3:53_1_1"><i><font color="Green" title="E34">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">k'</font> = <font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c12">a'</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E7:53_1_1"/>
for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c11">k'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">k'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c13">u</font></span><br/>

<a NAME="E1:53_1_1_1"/><i><font color="Green" title="E36">A7</font></i>: 
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c13">u</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E2:53_1_1_1"/>
 <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c13">u</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:53"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E3:53_1_1_1"/><b>then </b><i><font color="Green" title="E37">A8</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c13">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>, <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D3" title="WAYBEL_9:def.3">WAYBEL_9:def 3</a></i>;<br/>
<a NAME="E4:53_1_1_1"/><i><font color="Green" title="E38">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c13">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:53_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="waybel25.html#E34"><i><font color="Green" title="E28">Lm6</font></i></a></i>;<br/>
<i><font color="Green" title="E39">A10</font></i>: <font color="Maroon" title="c12">a'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c13">u</font> = 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c13">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E3:53"><i><font color="Green" title="E31">A2</font></i></a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1" title="WAYBEL_2:def.1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c13">u</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E6:53_1_1_1"/>
<font color="Maroon" title="c11">k'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c13">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:53_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="waybel25.html#T24" target="_self" title="WAYBEL25:th.24">Th24</a></i>;<br/>
<a NAME="E7:53_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">k'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c13">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:53_1_1_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:53_1_1_1"/>
<font color="Maroon" title="c11">k'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c13">u</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:53_1_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="waybel25.html#E3:53_1_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="waybel25.html#E5:53_1_1_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:53_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E2:53_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">b</font> implies <font color="Maroon" title="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">b</font> )</span><br/>

<b>assume </b><a NAME="E4:53_1"/><i><font color="Green" title="E34">A11</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> holds <br/><font color="Olive" title="b1">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">b</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c9">a'</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E6:53_1"/>
for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c9">a'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">a'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c11">u</font></span><br/>

<a NAME="E1:53_1_2"/><i><font color="Green" title="E35">A12</font></i>: 
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c11">u</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<i><font color="Green" title="E36">A13</font></i>: <font color="Maroon" title="c9">a'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c11">u</font> = 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E3:53"><i><font color="Green" title="E31">A2</font></i></a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1" title="WAYBEL_2:def.1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">u</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E3:53_1_2"/>
 <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">u</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:53"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E4:53_1_2"/><b>then </b><i><font color="Green" title="E37">A14</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>, <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D3" title="WAYBEL_9:def.3">WAYBEL_9:def 3</a></i>;<br/>
<a NAME="E5:53_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">b</font> <a href="yellow14.html#K1" title="YELLOW14:func.1">.</a> <font color="Maroon" title="c11">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> or <font color="Maroon" title="c12">s</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="yellow14.html#K1" title="YELLOW14:func.1">.</a> <font color="Maroon" title="c11">u</font> )</span><br/>

<b>assume </b><a NAME="E1:53_1_2_2"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">s</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="yellow14.html#K1" title="YELLOW14:func.1">.</a> <font color="Maroon" title="c11">u</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:53_1_2_2"/><i><font color="Green" title="E38">A15</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:53_1_2_2"/><i><font color="Green" title="E39">A16</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c13">i</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="waybel25.html#E1:53_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="waybel25.html#E3:53_1_2_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="waybel25.html#E34"><i><font color="Green" title="E28">Lm6</font></i></a></i>;<br/>
<a NAME="E6:53_1_2_2"/><i><font color="Green" title="E40">A17</font></i>: 
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">u</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:53_1_2_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="waybel25.html#T24" target="_self" title="WAYBEL25:th.24">Th24</a></i>;<br/>
<a NAME="E7:53_1_2_2"/><i><font color="Green" title="E41">A18</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:53_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:53_1_2_2"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">i</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel24.html#T21" title="WAYBEL24:th.21">WAYBEL24:21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">k</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19" title="WAYBEL24:th.19">WAYBEL24:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">k'</font> = <font color="Maroon" title="c15">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E11:53_1_2_2"/>
<font color="Maroon" title="c15">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:53_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="waybel25.html#E7:53_1_2_2"><i><font color="Green" title="E41">A18</font></i></a></i>;<br/>
<a NAME="E12:53_1_2_2"/><b>then </b>
<font color="Maroon" title="c16">k'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:53_1_2_2"/>
<font color="Maroon" title="c12">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="yellow14.html#K1" title="YELLOW14:func.1">.</a> <font color="Maroon" title="c11">u</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:53_1_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="waybel25.html#E6:53_1_2_2"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:53_1_2"/>
<font color="Maroon" title="c9">a'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c11">u</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:53_1_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="waybel25.html#E2:53_1_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="waybel25.html#E4:53_1_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:53_1"/>
<font color="Maroon" title="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E2:53_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:53"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T15" title="YELLOW_0:th.15">YELLOW_0:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
