<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:5_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E2:5_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:5_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:5_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:5_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:5_1_1_1"/>
( <font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 ;<br/><b>hence </b><a NAME="E6:5_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:5_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:5_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:5_1_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_1_2_1"/>
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span>
 ;<br/><a NAME="E2:5_1_1_2_1"/><b>then </b>
( not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font> &amp; not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:5_1_1_2_1"/><b>then </b>
( <font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:5_1_1_2_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E5:5_1_1_2_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:5_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E9:5_1_1"/><b>then </b>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E10:5_1_1"/><b>then </b>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="polynom1.html#D8">POLYNOM1:def 8</a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> = 
<font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<i><font color="Green">E47</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> = 
<font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_5_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> or  not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_5_1_1_1"><b>case </b></a><a NAME="E1:5_1_1_5_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_1_1_5_1_1_1"/>
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_5_1_1_2"><b>case </b></a><a NAME="E1:5_1_1_5_1_1_2"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:5_1_1_5_1_1_2"/><b>then </b>
( <font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:5_1_1_5_1_1_2"/>
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:5_1_1_5"/>
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:5_1_1"/>
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>  holds  <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span>
 ;<br/>


</div>
