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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<a NAME="E1:46"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:46"/>
( 2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">A</font></span>)</span> implies ( <font color="Maroon">A</font> is <a href="relat_1.html#V2">non-empty</a> &amp; not <font color="Maroon">A</font> is <a href="pencil_1.html#V13" target="_self">trivial-yielding</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:46_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">b1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:46_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:46_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i>, , <a class="txt" href="pencil_1.html#E1:46"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>thus </b><a NAME="E10:46_1"/>
<font color="Maroon">A</font> is <a href="relat_1.html#V2">non-empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:46_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>hence </b><a NAME="E2:46_1_1"/>
not <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E11:46_1"/>
not <font color="Maroon">A</font> is <a href="pencil_1.html#V13" target="_self">trivial-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D16" target="_self">PENCIL_1:def 16</a><br/>

<b>thus </b><a NAME="E1:46_1_2"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E3:46_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>

<a NAME="E2:46_1_2"/>
( <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 
<b>by </b><i>, <a class="txt" href="pencil_1.html#E1:46"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_1.html#E3:46_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E3:46_1_2"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E3:46_1"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:46_1_2"/>
not <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:46"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">A</font> is <a href="relat_1.html#V2">non-empty</a> &amp; not <font color="Maroon">A</font> is <a href="pencil_1.html#V13" target="_self">trivial-yielding</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><a NAME="E4:46_2"/>
contradiction
 <b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, , , <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:46"/><b>then </b>
not  <a href="card_3.html#K4">product</a> <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="card_3.html#T37">CARD_3:37</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:46"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:46"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T1" target="_self">Th1</a>, <a class="txt" href="pencil_1.html#E1:46"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:46"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">r</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:46"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:46"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:46"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">a</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">p</font>,<font color="Maroon">t</font>;<br/>
<a NAME="E19:46"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">p</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E1:46"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<i><font color="Green">E81</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">p</font>,<font color="Maroon">t</font></span>)</span> = 
<font color="Maroon">I</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">A</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="46_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_4_1"/>
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> or <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_4_1_1"><b>suppose </b></a><a NAME="E1:46_4_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:46_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">p</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:46_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">p</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:46"><i><font color="Green">E34</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_4_1_2"><b>suppose </b></a><a NAME="E1:46_4_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">p</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:46"><i><font color="Green">E34</font></i></a>, , , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E22:46"/><b>then </b>
<font color="Maroon">a</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">p</font>,<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T2" target="_self">Th2</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E23:46"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="pencil_1.html#E1:46"><i><font color="Green">E34</font></i></a>, , , </i>;<br/>


</div>
