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<b>let </b><font color="Maroon" title="c1">N</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<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>for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font>;<br/>
<a NAME="E1:7"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font> holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">e'</font> = <font color="Maroon" title="c2">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> ;<br/>
<b>consider </b><font color="Maroon" title="c4">u'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E4:7_1"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c3">e'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">u'</font> &amp; <font color="Maroon" title="c3">e'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">u'</font> )
 <b>by </b><i><a class="ref" href="waybel28.html#T5" target="_self" title="WAYBEL28:th.5">Th5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">u'</font>
;<br/>

<b>thus </b><a NAME="E5:7_1"/>
<font color="Maroon" title="c4">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">n</font>, <font color="Maroon" title="c6">m</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>;<br/>

<b>assume </b><a NAME="E6:7_1"/>
( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c4">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> )
 ;<br/>

<b>hence </b><a NAME="E7:7_1"/>
<font color="Maroon" title="c5">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E4:7_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">p</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:7"/><i><font color="Green" title="E8">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">p</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="c1">N</font>
 <b>and </b><br/><a NAME="E4:7"/><i><font color="Green" title="E9">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">p</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="c1">N</font>
 <b>and </b><br/><a NAME="E5:7"/><i><font color="Green" title="E10">A5</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="waybel28.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">p</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="waybel28.html#E3:7"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="waybel28.html#E4:7"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">p</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel28.html#D1" target="_self" title="WAYBEL28:def.1">WAYBEL28:def 1</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">j'</font> = <font color="Maroon" title="c4">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> ;<br/>
<b>thus </b><a NAME="E8:7"/>
<font color="Maroon" title="c4">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E5:7"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/>


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