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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ;<br/><b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">F</font> holds  <font color="Olive" title="b1">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</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 <font color="Maroon" title="c2">N</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">i</font></span>)</span> );<br/>
<div><i><font color="Green" title="E8">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">x</font> as    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font> <b>by </b><i><a class="txt" href="yellow19.html#E1:11"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E4:11_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow19.html#D2" target="_self" title="YELLOW19:def.2">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">i</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y</font>;<br/><b>thus </b><a NAME="E6:11_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>
 ;<br/><b>thus </b><a NAME="E7:11_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">y</font>]
 <b>by </b><i><a class="txt" href="yellow19.html#E4:11_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green" title="E9">A4</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> )
 <b>and </b><br/><a NAME="E5:11"/><i><font color="Green" title="E10">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="yellow19.html#E2:11"><i><font color="Green" title="E8">A2</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">B</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</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 <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="txt" href="yellow19.html#E4:11"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>
<a NAME="E7:11"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">N</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a>, <a class="ref" href="waybel_0.html#D6" title="WAYBEL_0:def.6">WAYBEL_0:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E9:11"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E10:11"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">C</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> as    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="yellow19.html#D2" target="_self" title="YELLOW19:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">C</font>
;<br/>

<b>let </b><font color="Maroon" title="c8">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><br/>

<b>assume </b><a NAME="E12:11"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:11"/><i><font color="Green" title="E12">A7</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</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/>
<a NAME="E15:11"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c9">y</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"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow19.html#E14:11"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> <b>by </b><i><a class="txt" href="yellow19.html#E14:11"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>
<a NAME="E17:11"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon" title="c6">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_9.html#T12" title="WAYBEL_9:th.12">WAYBEL_9:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E19:11"/><i><font color="Green" title="E14">A9</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c6">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="txt" href="yellow19.html#E15:11"><i><font color="Green" title="E13">A8</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:11_2"/><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E3:11_2"/><i><font color="Green" title="E16">A11</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow19.html#E5:11"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/><a NAME="E4:11_2"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E4:11"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="yellow19.html#E1:11_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="yellow19.html#E3:11_2"><i><font color="Green" title="E16">A11</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="E5:11_2"/><b>then </b>
<font color="Maroon" title="c13">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E10:11"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/><a NAME="E6:11_2"/><b>then </b>
<font color="Maroon" title="c13">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E19:11"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><a NAME="E7:11_2"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon" title="c13">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow19.html#E19:11"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_9.html#T12" title="WAYBEL_9:th.12">WAYBEL_9:12</a></i>;<br/><font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">y</font>
<b>by </b><i><a class="txt" href="yellow19.html#E14:11"><i><font color="Green" title="E12">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="txt" href="yellow19.html#E19:11"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="waybel_9.html#T16" title="WAYBEL_9:th.16">WAYBEL_9:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c14">z</font>
<b>by </b><i><a class="txt" href="yellow19.html#E19:11"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="waybel_9.html#T16" title="WAYBEL_9:th.16">WAYBEL_9:16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font>

;<br/><b>hence </b><a NAME="E10:11_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:11_2"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:11"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


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