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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</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">B</font> being <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>  <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L</font> holds   <a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">B</font></span>)</span> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a></span><br/><b>let </b><font color="Maroon" title="c2">B</font> be  <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>  <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a></span><br/>



<a NAME="E1:100"/><i><font color="Green" title="E55">A1</font></i>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="ref" href="waybel23.html#D8" target="_self" title="WAYBEL23:def.8">Def8</a></i>;<br/>
<a NAME="E2:100"/><b>then </b><i><font color="Green" title="E56">A2</font></i>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E3:100"/><i><font color="Green" title="E57">A3</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font> is <a href="yellow_0.html#V6" title="YELLOW_0:attr.6">join-inheriting</a>
 
<b>by </b><i><a class="ref" href="waybel23.html#D2" target="_self" title="WAYBEL23:def.2">Def2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c4">X</font></span><br/><b>reconsider </b><font color="Maroon" title="c5">supX</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">unionX</font> =  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel13.html#T3" title="WAYBEL13:th.3">WAYBEL13:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">J</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:100_1"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c5">supX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">J</font>
 <b>and </b><br/><a NAME="E5:100_1"/><i><font color="Green" title="E59">A5</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">supX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c7">J</font>
 <b>by </b><i><a class="ref" href="waybel23.html#D11" target="_self" title="WAYBEL23:def.11">Def11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">dfuX</font> =  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel13.html#T3" title="WAYBEL13:th.3">WAYBEL13:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">fuX</font> =  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel13.html#T3" title="WAYBEL13:th.3">WAYBEL13:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:100_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></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> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> )</span><br/><b>thus </b><a NAME="E2:100_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span></span><br/><a NAME="E3:100_1_1"/><i><font color="Green" title="E60">A6</font></i>: 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c8">dfuX</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c8">dfuX</font> )</span><br/>

<b>assume </b><a NAME="E1:100_1_1_1"/><i><font color="Green" title="E61">A7</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c8">dfuX</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>consider </b><font color="Maroon" title="c12">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:100_1_1_1"/><i><font color="Green" title="E62">A8</font></i>: 
<font color="Maroon" title="c12">y1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c11">x1</font>
 <b>and </b><br/><a NAME="E5:100_1_1_1"/><i><font color="Green" title="E63">A9</font></i>: 
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E1:100_1_1_1"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="waybel_0.html#D15" title="WAYBEL_0:def.15">WAYBEL_0:def 15</a></i>;<br/>
<a NAME="E6:100_1_1_1"/>
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">V</font>,<font color="Maroon" title="c1">L</font></span>)</span> where <font color="Olive" title="b1">V</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> :  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">V</font>,<font color="Maroon" title="c1">L</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel23.html#E5:100_1_1_1"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="waybel_0.html#D27" title="WAYBEL_0:def.27">WAYBEL_0:def 27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">Y</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:100_1_1_1"/><i><font color="Green" title="E64">A10</font></i>: 
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c13">Y</font>,<font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E9:100_1_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c13">Y</font>,<font color="Maroon" title="c1">L</font>
 ;<br/>
<b>defpred </b><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> ] <b>means </b> ex <font color="Olive" title="b1">I</font> being  <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#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span></span>)</span> ex <font color="Olive" title="b2">z1</font>, <font color="Olive" title="b3">z2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b3">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> &amp; <font color="Olive" title="b2">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">z2</font> );<br/>
<a NAME="E10:100_1_1_1"/><i><font color="Green" title="E65">A11</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> holds <br/> ex <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">v</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> implies  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">z</font>,<font color="Olive" title="b1">v</font>] )</span><br/>

<b>assume </b><a NAME="E1:100_1_1_1_1"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">z</font>,<font color="Olive" title="b1">v</font>]</span><br/>

<b>then consider </b><font color="Maroon" title="c15">J</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:100_1_1_1_1"/><i><font color="Green" title="E66">A12</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J</font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_1"/><i><font color="Green" title="E67">A13</font></i>: 
<font color="Maroon" title="c15">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">I</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:100_1_1_1_1"/>
<font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E7:100_1_1_1_1"/><i><font color="Green" title="E68">A14</font></i>: 
<font color="Maroon" title="c16">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>and </b><br/><a NAME="E8:100_1_1_1_1"/><i><font color="Green" title="E69">A15</font></i>: 
<font color="Maroon" title="c15">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">I</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E4:100_1_1_1_1"><i><font color="Green" title="E67">A13</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">I</font> = <font color="Maroon" title="c16">I</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#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel23.html#E7:100_1_1_1_1"><i><font color="Green" title="E68">A14</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">I1</font> = <font color="Maroon" title="c17">I</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/>
<a NAME="E11:100_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">I</font> is   <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#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">J</font> = <font color="Maroon" title="c15">J</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#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel23.html#E8:100_1_1_1_1"><i><font color="Green" title="E69">A15</font></i></a></i>;<br/>
<a NAME="E13:100_1_1_1_1"/>
<font color="Maroon" title="c19">J</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">z1</font> = <font color="Maroon" title="c14">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel23.html#E3:100_1_1_1_1"><i><font color="Green" title="E66">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">I2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E16:100_1_1_1_1"/><i><font color="Green" title="E70">A16</font></i>: 
<font color="Maroon" title="c18">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">I2</font>
 <b>and </b><br/><a NAME="E17:100_1_1_1_1"/><i><font color="Green" title="E71">A17</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c21">I2</font>
 <b>by </b><i><a class="ref" href="waybel23.html#D11" target="_self" title="WAYBEL23:def.11">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E19:100_1_1_1_1"/><i><font color="Green" title="E72">A18</font></i>: 
<font color="Maroon" title="c22">z2</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c20">z1</font>
 <b>and </b><br/><a NAME="E20:100_1_1_1_1"/><i><font color="Green" title="E73">A19</font></i>: 
<font color="Maroon" title="c22">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">I2</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E3:100_1_1_1_1"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_1_1"><i><font color="Green" title="E69">A15</font></i></a>, <a class="txt" href="waybel23.html#E17:100_1_1_1_1"><i><font color="Green" title="E71">A17</font></i></a>, <a class="ref" href="waybel_0.html#D15" title="WAYBEL_0:def.15">WAYBEL_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">v</font> = <font color="Maroon" title="c22">z2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font> <b>by </b><i><a class="txt" href="waybel23.html#E16:100_1_1_1_1"><i><font color="Green" title="E70">A16</font></i></a>, <a class="txt" href="waybel23.html#E20:100_1_1_1_1"><i><font color="Green" title="E73">A19</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c23">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c23">v</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c17">I</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</font> &amp; <font color="Olive" title="b2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">v</font> &amp; <font color="Maroon" title="c17">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">I</font> &amp; <font color="Olive" title="b1">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">z2</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c20">z1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c20">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</font> &amp; <font color="Olive" title="b1">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">v</font> &amp; <font color="Maroon" title="c17">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">I</font> &amp; <font color="Maroon" title="c20">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z2</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c22">z2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</font> &amp; <font color="Maroon" title="c22">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">v</font> &amp; <font color="Maroon" title="c17">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">I</font> &amp; <font color="Maroon" title="c20">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c22">z2</font> )</span><br/>

<b>thus </b><a NAME="E22:100_1_1_1_1"/>
( <font color="Maroon" title="c20">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</font> &amp; <font color="Maroon" title="c22">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">v</font> &amp; <font color="Maroon" title="c17">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">I</font> &amp; <font color="Maroon" title="c20">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c22">z2</font> )
 <b>by </b><i><a class="txt" href="waybel23.html#E7:100_1_1_1_1"><i><font color="Green" title="E68">A14</font></i></a>, <a class="txt" href="waybel23.html#E16:100_1_1_1_1"><i><font color="Green" title="E70">A16</font></i></a>, <a class="txt" href="waybel23.html#E19:100_1_1_1_1"><i><font color="Green" title="E72">A18</font></i></a>, <a class="txt" href="waybel23.html#E20:100_1_1_1_1"><i><font color="Green" title="E73">A19</font></i></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>consider </b><font color="Maroon" title="c14">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">Y</font>,<font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E12:100_1_1_1"/><i><font color="Green" title="E66">A20</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>]
 <b>from </b><i><a class="ref" href="monoid_1.html#S1" title="MONOID_1:sch.1">MONOID_1:sch 1</a>(<a class="txt" href="waybel23.html#E10:100_1_1_1"><i><font color="Green" title="E65">A11</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c15">srg</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/>
<a NAME="E14:100_1_1_1"/><i><font color="Green" title="E67">A21</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Y</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/>
<b>reconsider </b><font color="Maroon" title="c16">Z</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E16:100_1_1_1"/><i><font color="Green" title="E68">A22</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c16">Z</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_1_1_1_2_1"/>
(  not <font color="Maroon" title="c16">Z</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c16">Z</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1_2_1_1"/>
not <font color="Maroon" title="c16">Z</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <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> <font color="Maroon" title="c16">Z</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><b>hence </b><a NAME="E2:100_1_1_1_2_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c16">Z</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T54" title="YELLOW_0:th.54">YELLOW_0:54</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_1_2_1_2"/>
<font color="Maroon" title="c16">Z</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <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> <font color="Maroon" title="c16">Z</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span><br/><div class="add"><b>hence </b><a NAME="E2:100_1_1_1_2_1_2"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c16">Z</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T42" title="YELLOW_0:th.42">YELLOW_0:42</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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E17:100_1_1_1"/><i><font color="Green" title="E69">A23</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E18:100_1_1_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_1_1_1_3_1"/>
(  not <font color="Maroon" title="c16">Z</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c16">Z</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1_3_1_1"/>
not <font color="Maroon" title="c16">Z</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:100_1_1_1_3_1_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E3:100"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="waybel21.html#T15" title="WAYBEL21:th.15">WAYBEL21:15</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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_1_3_1_2"/>
<font color="Maroon" title="c16">Z</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:100_1_1_1_3_1_2"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E1:100"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 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><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c17">xl</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</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_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> ;<br/>
<a NAME="E20:100_1_1_1"/><i><font color="Green" title="E70">A24</font></i>: 
(  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c16">Z</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 <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c16">Z</font> holds <br/> <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font> ) )
 
<b>by </b><i><a class="txt" href="waybel23.html#E17:100_1_1_1"><i><font color="Green" title="E69">A23</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/>
<a NAME="E21:100_1_1_1"/><i><font color="Green" title="E71">A25</font></i>: 
<font color="Maroon" title="c17">xl</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c16">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</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="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font> or <font color="Maroon" title="c18">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c17">xl</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c19">b1</font> = <font color="Maroon" title="c18">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/>
<b>assume </b><a NAME="E2:100_1_1_1_4"/>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c17">xl</font></span><br/>

<a NAME="E3:100_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c19">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E20:100_1_1_1"><i><font color="Green" title="E70">A24</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:100_1_1_1_4"/>
<font color="Maroon" title="c18">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">xl</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></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="c18">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c16">Z</font> implies <font color="Maroon" title="c17">xl</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">b</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c19">b1</font> = <font color="Maroon" title="c18">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><b>assume </b><a NAME="E2:100_1_1_1_5"/><i><font color="Green" title="E72">A26</font></i>: 
<font color="Maroon" title="c18">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c16">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">xl</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">b</font></span><br/><a NAME="E3:100_1_1_1_5"/>
<font color="Maroon" title="c19">b1</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c16">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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"> (  not <font color="Maroon" title="c20">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font> or <font color="Maroon" title="c20">c1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">b1</font> )</span><br/>

<b>assume </b><a NAME="E1:100_1_1_1_5_1"/><i><font color="Green" title="E73">A27</font></i>: 
<font color="Maroon" title="c20">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">c1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">b1</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c21">c</font> = <font color="Maroon" title="c20">c1</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_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> ;<br/>
<a NAME="E3:100_1_1_1_5_1"/>
<font color="Maroon" title="c21">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">b</font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E2:100_1_1_1_5"><i><font color="Green" title="E72">A26</font></i></a>, <a class="txt" href="waybel23.html#E1:100_1_1_1_5_1"><i><font color="Green" title="E73">A27</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:100_1_1_1_5_1"/>
<font color="Maroon" title="c20">c1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">b1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</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><a NAME="E4:100_1_1_1_5"/><b>then </b>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">b1</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E17:100_1_1_1"><i><font color="Green" title="E69">A23</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/><b>hence </b><a NAME="E5:100_1_1_1_5"/>
<font color="Maroon" title="c17">xl</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</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>
<a NAME="E23:100_1_1_1"/><b>then </b><i><font color="Green" title="E72">A28</font></i>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E21:100_1_1_1"><i><font color="Green" title="E71">A25</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/>
<a NAME="E24:100_1_1_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c13">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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"> (  not <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> or <font color="Maroon" title="c18">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:100_1_1_1_6"/><i><font color="Green" title="E73">A29</font></i>: 
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c1">L</font></span><br/>

<b>then consider </b><font color="Maroon" title="c19">I</font> being   <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#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon" title="c20">a1</font>, <font color="Maroon" title="c21">a2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:100_1_1_1_6"/><i><font color="Green" title="E74">A30</font></i>: 
<font color="Maroon" title="c20">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a</font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_6"/><i><font color="Green" title="E75">A31</font></i>: 
<font color="Maroon" title="c21">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font>
 <b>and </b><br/><a NAME="E5:100_1_1_1_6"/>
<font color="Maroon" title="c19">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>and </b><br/><a NAME="E6:100_1_1_1_6"/>
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">I</font>
 <b>and </b><br/><a NAME="E7:100_1_1_1_6"/><i><font color="Green" title="E76">A32</font></i>: 
<font color="Maroon" title="c20">a1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">a2</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E12:100_1_1_1"><i><font color="Green" title="E66">A20</font></i></a></i>;<br/>
<a NAME="E8:100_1_1_1_6"/><i><font color="Green" title="E77">A33</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</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> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E14:100_1_1_1"><i><font color="Green" title="E67">A21</font></i></a>, <a class="txt" href="waybel23.html#E1:100_1_1_1_6"><i><font color="Green" title="E73">A29</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="E9:100_1_1_1_6"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<a NAME="E10:100_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c21">a2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E4:100_1_1_1_6"><i><font color="Green" title="E75">A31</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_1_6"><i><font color="Green" title="E77">A33</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E11:100_1_1_1_6"/>
<font color="Maroon" title="c18">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E3:100_1_1_1_6"><i><font color="Green" title="E74">A30</font></i></a>, <a class="txt" href="waybel23.html#E7:100_1_1_1_6"><i><font color="Green" title="E76">A32</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</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>
<a NAME="E25:100_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c13">Y</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">srg</font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E23:100_1_1_1"><i><font color="Green" title="E72">A28</font></i></a>, <a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<a NAME="E26:100_1_1_1"/><b>then </b><i><font color="Green" title="E73">A34</font></i>: 
<font color="Maroon" title="c11">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">srg</font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E4:100_1_1_1"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_1"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E27:100_1_1_1"/><i><font color="Green" title="E74">A35</font></i>: 
 <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></i>;<br/>
<a NAME="E28:100_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c18">a</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> <font color="Maroon" title="c14">g</font> or <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<b>assume </b><a NAME="E1:100_1_1_1_7"/>
<font color="Maroon" title="c18">a</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> <font color="Maroon" title="c14">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span><br/>

<b>then consider </b><font color="Maroon" title="c19">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:100_1_1_1_7"/><i><font color="Green" title="E75">A36</font></i>: 
<font color="Maroon" title="c19">b</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> <font color="Maroon" title="c14">g</font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_7"/><i><font color="Green" title="E76">A37</font></i>: 
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">b</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>consider </b><font color="Maroon" title="c20">I</font> being   <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#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon" title="c21">b1</font>, <font color="Maroon" title="c22">b2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:100_1_1_1_7"/>
<font color="Maroon" title="c21">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font>
 <b>and </b><br/><a NAME="E7:100_1_1_1_7"/>
<font color="Maroon" title="c22">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">b</font>
 <b>and </b><br/><a NAME="E8:100_1_1_1_7"/><i><font color="Green" title="E77">A38</font></i>: 
<font color="Maroon" title="c20">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>and </b><br/><a NAME="E9:100_1_1_1_7"/><i><font color="Green" title="E78">A39</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">I</font>
 <b>and </b><br/><a NAME="E10:100_1_1_1_7"/>
<font color="Maroon" title="c21">b1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c22">b2</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E12:100_1_1_1"><i><font color="Green" title="E66">A20</font></i></a>, <a class="txt" href="waybel23.html#E3:100_1_1_1_7"><i><font color="Green" title="E75">A36</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:100_1_1_1_7"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E4:100_1_1_1_7"><i><font color="Green" title="E76">A37</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_1_7"><i><font color="Green" title="E77">A38</font></i></a>, <a class="txt" href="waybel23.html#E9:100_1_1_1_7"><i><font color="Green" title="E78">A39</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
<a NAME="E29:100_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c16">Z</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">V</font>,<span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> where <font color="Olive" title="b1">V</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span>)</span> :  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">V</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel23.html#E16:100_1_1_1"><i><font color="Green" title="E68">A22</font></i></a></i>;<br/>
<a NAME="E30:100_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D27" title="WAYBEL_0:def.27">WAYBEL_0:def 27</a></i>;<br/>
<b>hence </b><a NAME="E31:100_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c8">dfuX</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E26:100_1_1_1"><i><font color="Green" title="E73">A34</font></i></a>, <a class="txt" href="waybel23.html#E27:100_1_1_1"><i><font color="Green" title="E74">A35</font></i></a>, <a class="ref" href="waybel_0.html#D15" title="WAYBEL_0:def.15">WAYBEL_0:def 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><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:100_1_1_2"/><i><font color="Green" title="E61">A40</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span><br/><a NAME="E2:100_1_1_2"/><b>then </b><i><font color="Green" title="E62">A41</font></i>: 
<font color="Maroon" title="c10">x</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">x</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <b>by </b><i><a class="txt" href="waybel23.html#E1:100_1_1_2"><i><font color="Green" title="E61">A40</font></i></a>, <a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">x2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:100_1_1_2"/><i><font color="Green" title="E63">A42</font></i>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font>
 <b>and </b><br/><a NAME="E6:100_1_1_2"/><i><font color="Green" title="E64">A43</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c12">x2</font>
 <b>by </b><i><a class="ref" href="waybel23.html#D11" target="_self" title="WAYBEL23:def.11">Def11</a></i>;<br/><a NAME="E7:100_1_1_2"/>
<font color="Maroon" title="c11">x1</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> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E2:100_1_1_2"><i><font color="Green" title="E62">A41</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:100_1_1_2"/><b>then </b><i><font color="Green" title="E65">A44</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E1:100_1_1_2"><i><font color="Green" title="E61">A40</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>thus </b><a NAME="E9:100_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> or <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:100_1_1_2_1"/><i><font color="Green" title="E66">A45</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span><br/>

<a NAME="E2:100_1_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c12">x2</font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E5:100_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></i>;<br/>
<b>hence </b><a NAME="E3:100_1_1_2_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E6:100_1_1_2"><i><font color="Green" title="E64">A43</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_2"><i><font color="Green" title="E65">A44</font></i></a>, <a class="txt" href="waybel23.html#E1:100_1_1_2_1"><i><font color="Green" title="E66">A45</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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></div><b>end;</b></div><a NAME="E5:100_1_1"/><b>then </b>
 <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T94" title="ZFMISC_1:th.94">ZFMISC_1:94</a></i>;<br/><a NAME="E6:100_1_1"/><b>then </b><i><font color="Green" title="E61">A46</font></i>: 
 <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c6">unionX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel23.html#T2" target="_self" title="WAYBEL23:th.2">Th2</a></i>;<br/><a NAME="E7:100_1_1"/>
 <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <font color="Maroon" title="c6">unionX</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E2:100"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="waybel23.html#E3:100"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="waybel23.html#T5" target="_self" title="WAYBEL23:th.5">Th5</a></i>;<br/><a NAME="E8:100_1_1"/><b>then </b>
 <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E6:100_1_1"><i><font color="Green" title="E61">A46</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:100_1_1"/><b>then </b><i><font color="Green" title="E62">A47</font></i>: 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c9">fuX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_3.html#T6" title="YELLOW_3:th.6">YELLOW_3:6</a></i>;<br/><a NAME="E10:100_1_1"/>
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c9">fuX</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T11" target="_self" title="WAYBEL23:th.11">Th11</a></i>;<br/><a NAME="E11:100_1_1"/><b>then </b>
<font color="Maroon" title="c8">dfuX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E9:100_1_1"><i><font color="Green" title="E62">A47</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:100_1_1"/><b>then </b>
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c8">dfuX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p3">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p0">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_3.html#T6" title="YELLOW_3:th.6">YELLOW_3:6</a></i>;<br/><a NAME="E13:100_1_1"/><b>then </b><i><font color="Green" title="E63">A48</font></i>: 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c8">dfuX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel23.html#T7" target="_self" title="WAYBEL23:th.7">Th7</a></i>;<br/><b>thus </b> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span> = 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" title="WAYBEL_0:func.11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" title="WAYBEL23:func.1">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel23.html#T6" target="_self" title="WAYBEL23:th.6">Th6</a></i>
<br/>.= 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c8">dfuX</font>
<b>by </b><i><a class="txt" href="waybel23.html#E3:100_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="waybel23.html#E13:100_1_1"><i><font color="Green" title="E63">A48</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel23.html#E4:100_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="waybel23.html#E5:100_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="waybel23.html#T6" target="_self" title="WAYBEL23:th.6">Th6</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="E9:100_1"/>
 <a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</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="E5:100"/>
 <a href="waybel23.html#K4" title="WAYBEL23:func.4">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">B</font></span>)</span> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D33" title="WAYBEL_0:def.33">WAYBEL_0:def 33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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