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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">N</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> ; <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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span>, <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b2">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) holds <br/><font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b4">b</font> ) &amp; ( for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">p'</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b6">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b5">b</font> iff <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b5">b</font> ) ) holds <br/>( <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b5">b</font> iff <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p'</font>,<font color="Olive" title="b5">b</font> ) ) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/>for <font color="Olive" title="b3">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b5">b</font> iff <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p</font>,<font color="Olive" title="b5">b</font> )</span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span>, <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b3">b</font> ) holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">b</font> ) &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p'</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> iff <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) ) holds <br/>( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">b</font> iff <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p'</font>,<font color="Olive" title="b4">b</font> ) ) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 implies  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/>for <font color="Olive" title="b2">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b4">b</font> iff <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b4">b</font> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green" title="E18">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b3">b</font> ) holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">b</font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green" title="E19">A2</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p'</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> iff <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) ) holds <br/>( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">b</font> iff <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p'</font>,<font color="Olive" title="b4">b</font> )
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 ; <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">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/>for <font color="Olive" title="b2">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b4">b</font> iff <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b4">b</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c4">o</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>] <b>means </b>( $2 <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> $1,$3 &amp; ( $2 <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $1,$3 or $2 <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c4">o</font>,$3 ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font>] <b>means </b>for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] iff <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> $2,<font color="Olive" title="b2">b</font> );<br/>
<a NAME="E5:24"/><i><font color="Green" title="E21">A4</font></i>: 
for <font color="Olive" title="b1">o'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>  ex <font color="Olive" title="b2">o1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">o'</font>,<font color="Olive" title="b2">o1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">o'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</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">o1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">o'</font>,<font color="Olive" title="b1">o1</font>]</span><br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>] <b>means </b><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">o'</font>,$1,$2];<br/>
<b>consider </b><font color="Maroon" title="c6">o1</font> being   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E2:24_1"/><i><font color="Green" title="E22">A5</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] )
 <b>from </b><i><a class="ref" href="relset_1.html#S2" title="RELSET_1:sch.2">RELSET_1:sch 2</a>();<br/></i>
<div><i><font color="Green" title="E23">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> )</span><br/><a NAME="E1:24_1_1"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font>] )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E2:24_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:24_1"><i><font color="Green" title="E22">A5</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> )</span><br/><a NAME="E1:24_1_2"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font>] )
 <b>by </b><i><a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E2:24_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o1</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:24_1"><i><font color="Green" title="E22">A5</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>then reconsider </b><font color="Maroon" title="c7">o1</font> = <font color="Maroon" title="c6">o1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="arrow.html#E3:24_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="arrow.html#D1" target="_self" title="ARROW:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c8">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o1</font> implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )</span><br/><a NAME="E1:24_1_3"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c8">a</font>] implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="ref" href="arrow.html#T6" target="_self" title="ARROW:th.6">Th6</a></i>;<br/><b>hence </b><a NAME="E2:24_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c8">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o1</font> implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:24_1"><i><font color="Green" title="E22">A5</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>then reconsider </b><font color="Maroon" title="c8">o1</font> = <font color="Maroon" title="c7">o1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="arrow.html#D2" target="_self" title="ARROW:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">o1</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>2</sub></font>[<font color="Maroon" title="c5">o'</font>,<font color="Maroon" title="c8">o1</font>]</span><br/>

<b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</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="c5">o'</font>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>] iff <font color="Maroon" title="c9">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c10">b</font> )</span><br/>



<a NAME="E8:24_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">o1</font> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>] )
 
<b>by </b><i><a class="txt" href="arrow.html#E2:24_1"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:24_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">o'</font>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>] iff <font color="Maroon" title="c9">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c8">o1</font>,<font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="ref" href="arrow.html#D4" target="_self" title="ARROW:def.4">Def4</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>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>,   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span>] <b>means </b>for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>2</sub></font>[$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,$2 <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b1">i</font>];<br/>
<a NAME="E6:24"/><i><font color="Green" title="E22">A7</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">p'</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">p</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b3">p'</font>] holds <br/><font color="Olive" title="b2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">p'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p'</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b2">p'</font>] holds <br/><font color="Olive" title="b1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">p'</font></span><br/><b>let </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">p'</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</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">S<sub>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">p</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">p'</font>] implies <font color="Maroon" title="c6">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">p'</font> )</span><br/>





<b>assume </b><a NAME="E1:24_2"/><i><font color="Green" title="E23">A8</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">p</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">p'</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">p'</font></span><br/>

<b>let </b><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D9" title="FUNCT_2:def.9">FUNCT_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c9">pi</font> = <font color="Maroon" title="c6">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="arrow.html#D1" target="_self" title="ARROW:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">pi'</font> = <font color="Maroon" title="c7">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="arrow.html#D1" target="_self" title="ARROW:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font> )</span><br/><a NAME="E1:24_2_1"/>
( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>] implies <font color="Maroon" title="c11">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c6">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c12">b</font> ) &amp; ( <font color="Maroon" title="c11">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c6">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c12">b</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>] implies <font color="Maroon" title="c11">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c7">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c12">b</font> ) &amp; ( <font color="Maroon" title="c11">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c7">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c12">b</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>] ) )
 <b>by </b><i><a class="txt" href="arrow.html#E1:24_2"><i><font color="Green" title="E23">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E2:24_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="ref" href="arrow.html#D4" target="_self" title="ARROW:def.4">Def4</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="E5:24_2"/><b>then </b>
<font color="Maroon" title="c9">pi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">pi'</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#D3" title="RELSET_1:def.3">RELSET_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:24_2"/>
<font color="Maroon" title="c6">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c8">i</font>
 ; <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="E7:24"/><i><font color="Green" title="E23">A9</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>  ex <font color="Olive" title="b2">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>; <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">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">p</font>]</span><br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font>] <b>means </b><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,$2];<br/>
<a NAME="E1:24_3"/><i><font color="Green" title="E24">A10</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">o1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">o1</font>]
 
<b>by </b><i><a class="txt" href="arrow.html#E5:24"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">p</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">N</font>, <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:24_3"/><i><font color="Green" title="E25">A11</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="arrow.html#E1:24_3"><i><font color="Green" title="E24">A10</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c6">p</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">p</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>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">p</font>]</span><br/>

<b>thus </b><a NAME="E5:24_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">p</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E3:24_3"><i><font color="Green" title="E25">A11</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>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>] <b>means </b> ex <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span> st <br/>( <font color="Maroon">S<sub>3</sub></font>[$1,<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 );<br/>
<a NAME="E8:24"/><i><font color="Green" title="E24">A12</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>  ex <font color="Olive" title="b2">o'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">o'</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>; <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">o'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">o'</font>]</span><br/>

<b>consider </b><font color="Maroon" title="c6">p</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E2:24_4"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">p</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E7:24"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</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>4</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font>]</span><br/>

<b>thus </b><a NAME="E3:24_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E2:24_4"><i><font color="Green" title="E25">A13</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="c5">f'</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>, <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E10:24"/><i><font color="Green" title="E25">A14</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="arrow.html#E8:24"><i><font color="Green" title="E24">A12</font></i></a>);<br/></i>
<a NAME="E11:24"/><i><font color="Green" title="E26">A15</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b3">b</font> ) holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b3">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">q</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">b</font> ) holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font>,<font color="Olive" title="b2">b</font></span><br/><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Maroon" title="c7">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">b</font> ) implies <font color="Maroon" title="c7">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">b</font> )</span><br/>





<b>assume </b><a NAME="E1:24_5"/><i><font color="Green" title="E27">A16</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Maroon" title="c7">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">b</font></span><br/>

<b>consider </b><font color="Maroon" title="c9">p</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E3:24_5"/><i><font color="Green" title="E28">A17</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c9">p</font>] &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:24"><i><font color="Green" title="E25">A14</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c9">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c8">b</font></span><br/><a NAME="E1:24_5_1"/>
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font>]
 <b>by </b><i><a class="txt" href="arrow.html#E1:24_5"><i><font color="Green" title="E27">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E2:24_5_1"/>
<font color="Maroon" title="c7">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c9">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E3:24_5"><i><font color="Green" title="E28">A17</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>hence </b><a NAME="E5:24_5"/>
<font color="Maroon" title="c7">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E1:24"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="arrow.html#E3:24_5"><i><font color="Green" title="E28">A17</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">q'</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">b</font> implies <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">b</font> implies <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b2">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">a</font> implies <font color="Olive" title="b2">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">a</font> ) &amp; ( <font color="Olive" title="b2">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">a</font> implies <font color="Olive" title="b2">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">a</font> ) ) ) holds <br/>( <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font>,<font color="Olive" title="b2">b</font> iff <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q'</font>,<font color="Olive" title="b2">b</font> )</span><br/><b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> implies <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> ) &amp; ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> implies <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> ) &amp; ( <font color="Maroon" title="c9">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c9">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">a</font> ) &amp; ( <font color="Maroon" title="c9">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c9">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">a</font> ) ) ) implies ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c9">b</font> iff <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q'</font>,<font color="Maroon" title="c9">b</font> ) )</span><br/><b>assume </b><a NAME="E1:24_6"/><i><font color="Green" title="E27">A18</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> implies <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> ) &amp; ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> implies <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> ) &amp; ( <font color="Maroon" title="c9">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c9">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">a</font> ) &amp; ( <font color="Maroon" title="c9">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c9">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">a</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c9">b</font> iff <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q'</font>,<font color="Maroon" title="c9">b</font> )</span><br/><b>consider </b><font color="Maroon" title="c10">p</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E3:24_6"/><i><font color="Green" title="E28">A19</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c10">p</font>] &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:24"><i><font color="Green" title="E25">A14</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">p'</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E5:24_6"/><i><font color="Green" title="E29">A20</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">q'</font>,<font color="Maroon" title="c11">p'</font>] &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q'</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:24"><i><font color="Green" title="E25">A14</font></i></a></i>;<br/><a NAME="E6:24_6"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c10">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> iff <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c10">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c9">b</font> iff <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c9">b</font> )</span><br/>

<a NAME="E1:24_6_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c8">a</font>] iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">q'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c8">a</font>] )
 
<b>by </b><i><a class="txt" href="arrow.html#E1:24_6"><i><font color="Green" title="E27">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:24_6_1"/>
( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c10">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c9">b</font> iff <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">p'</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E3:24_6"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="arrow.html#E5:24_6"><i><font color="Green" title="E29">A20</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>hence </b><a NAME="E7:24_6"/>
( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c9">b</font> iff <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q'</font>,<font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E2:24"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="arrow.html#E3:24_6"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="arrow.html#E5:24_6"><i><font color="Green" title="E29">A20</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>then consider </b><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E14:24"/><i><font color="Green" title="E27">A21</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font>,<font color="Olive" title="b3">b</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b3">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E3:24"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="arrow.html#E11:24"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="arrow.html#T14" target="_self" title="ARROW:th.14">Th14</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Olive" title="b3">b</font> iff <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">b</font> )</span><br/>

<b>let </b><font color="Maroon" title="c7">p</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Olive" title="b2">b</font> iff <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>,<font color="Olive" title="b2">b</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_7"><b>now </b></a><div class="add"><a NAME="E1:24_7"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E2:24_7"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c7">p</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E4:24_7"/><i><font color="Green" title="E28">A22</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">p</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</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>2</sub></font>[<font color="Maroon" title="c8">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c7">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c9">i</font>]</span><br/><b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</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="c8">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>] iff <font color="Maroon" title="c10">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c7">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">b</font> )</span><br/>





<a NAME="E1:24_7_1"/>
( <font color="Maroon" title="c10">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">b</font> or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> or <font color="Maroon" title="c10">a</font> <a href="arrow.html#NR4" title="ARROW:NR.4">&gt;_</a> <font color="Maroon" title="c7">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">b</font> )
 
<b>by </b><i><a class="ref" href="arrow.html#T6" target="_self" title="ARROW:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E2:24_7_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>] iff <font color="Maroon" title="c10">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c7">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">b</font> )
 <b>by </b><i><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</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="c9">p'</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K3" title="ARROW:func.3">LinOrders</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><br/><a NAME="E6:24_7"/><i><font color="Green" title="E29">A23</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c9">p'</font>] &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">q</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E10:24"><i><font color="Green" title="E25">A14</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</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">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c11">b</font> implies <font color="Maroon" title="c10">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c11">b</font> )</span><br/><b>assume </b><a NAME="E7:24_7"/>
<font color="Maroon" title="c10">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c11">b</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">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c11">b</font></span><br/><a NAME="E8:24_7"/><b>then </b>
( <font color="Maroon" title="c10">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c11">b</font> &amp; <font color="Maroon" title="c7">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">p'</font> )
 <b>by </b><i><a class="txt" href="arrow.html#E6:24"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="arrow.html#E14:24"><i><font color="Green" title="E27">A21</font></i></a>, <a class="txt" href="arrow.html#E4:24_7"><i><font color="Green" title="E28">A22</font></i></a>, <a class="txt" href="arrow.html#E6:24_7"><i><font color="Green" title="E29">A23</font></i></a></i>;<br/><b>hence </b><a NAME="E9:24_7"/>
<font color="Maroon" title="c10">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c11">b</font>
 <b>by </b><i><a class="txt" href="arrow.html#E6:24_7"><i><font color="Green" title="E29">A23</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>hence </b><a NAME="E16:24"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">p</font> <a href="arrow.html#K1" title="ARROW:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Olive" title="b2">b</font> iff <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>,<font color="Olive" title="b2">b</font> )
 <b>by </b><i><a class="ref" href="arrow.html#T13" target="_self" title="ARROW:th.13">Th13</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>
