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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K5">NAT</a>  st ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) holds <br/> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K48">multnat</a>  <a href="setwiseo.html#K7">$$</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Olive">b<sub>1</sub></font>,1</span>)</span>;<br/>
<a NAME="E1:141"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">c<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:141_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:141_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><a NAME="E3:141_1_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></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="141_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:141_1_1_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E2:141_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_1"><i><font color="Green">E78</font></i></a></i>;<br/><b>hence </b><a NAME="E3:141_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_1_1_1_1"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:141_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E2:141_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T65" target="_self">Th65</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:141_1"/><b>then </b>
 <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
1
<b>by </b><i><a class="ref" href="card_1.html#T50">CARD_1:50</a></i>
<br/>.= 
<a href="binop_2.html#K48">multnat</a>  <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#T10">BINOP_2:10</a>, <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E2:141"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="setwiseo.html#K2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:141_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:141_2"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:141_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T2" target="_self">Th2</a></i>;<br/>
<b>then </b><i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,1
<b>by </b><i><a class="ref" href="funcop_1.html#T35">FUNCOP_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="binop_2.html#D23">BINOP_2:def 23</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0;<br/>
<a NAME="E5:141_2"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:141_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:141_2_2"/><i><font color="Green">E83</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_2_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_2_1_1"><b>suppose </b></a><a NAME="E1:141_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:141_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="polynom1.html#E5:141_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_2_1_2"><b>suppose </b></a><a NAME="E1:141_2_2_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:141_2_2_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:141_2_2_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:141_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="polynom1.html#E1:141_2_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="polynom1.html#E2:141_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="polynom1.html#E2:141_2_2_1_2"><i><font color="Green">E85</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:141_2"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" target="_self">NatMinor</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K48">multnat</a>  <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E1:141_2"><i><font color="Green">E79</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="polynom1.html#K18" target="_self">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_4.html#T1">CARD_4:1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E9:141_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E1:141_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_7.html#T94">FUNCT_7:94</a></i>;<br/>
<a NAME="E10:141_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span>,1</span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T36">FUNCOP_1:36</a></i>;<br/>
<a NAME="E11:141_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<a href="binop_2.html#K48">multnat</a>  <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K48">multnat</a>  <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="setwop_2.html#T9">SETWOP_2:9</a></i>;<br/>
<a NAME="E12:141_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E14:141_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2_3"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E1:141_2_3"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> holds <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="polynom1.html#K18" target="_self">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> ;<br/>
<a NAME="E2:141_2_3_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:141_2_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="polynom1.html#E2:141_2_3_1"><i><font color="Green">E87</font></i></a></i>;<br/>
<a NAME="E5:141_2_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K18" target="_self">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span>
 
;<br/>
<a NAME="E6:141_2_3_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<a NAME="E7:141_2_3_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font> as  <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:141_2_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:141_2_3_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_2_3_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:141_2_3_1_1_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:141_2_3_1_1_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E6:141_2_3_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:141_2_3_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><b>hence </b><a NAME="E4:141_2_3_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_1_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="polynom1.html#E2:141_2_3_1_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:141_2_3_1_1_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:141_2_3_1_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:141_2_3_1_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E6:141_2_3_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:141_2_3_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self">Def17</a></i>;<br/><b>hence </b><a NAME="E5:141_2_3_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_1_1_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="polynom1.html#E2:141_2_3_1_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E10:141_2_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D17" target="_self">Def17</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:141_2_3"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> holds <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="funct_7.html#S1">FUNCT_7:sch 1</a>(<a class="txt" href="polynom1.html#E1:141_2_3"><i><font color="Green">E86</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E4:141_2_3"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:141_2_3_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:141_2_3_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E4:141_2_3_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="polynom1.html#E3:141_2_3_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:141_2_3_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E7:141_2_3_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="polynom1.html#E6:141_2_3_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="polynom1.html#K18" target="_self">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="polynom1.html#E3:141_2_3_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="polynom1.html#K18" target="_self">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="polynom1.html#E6:141_2_3_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green">E93</font></i>: <font color="Maroon">c<sub>16</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="polynom1.html#E3:141_2_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="polynom1.html#E4:141_2_3_2"><i><font color="Green">E90</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="polynom1.html#E3:141_2_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="polynom1.html#E1:141_2_3_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="polynom1.html#E3:141_2_3_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="polynom1.html#E6:141_2_3_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="polynom1.html#E7:141_2_3_2"><i><font color="Green">E92</font></i></a></i>
;<br/>
<a NAME="E11:141_2_3_2"/><i><font color="Green">E94</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<a NAME="E12:141_2_3_2"/><i><font color="Green">E95</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom1.html#E11:141_2_3_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green">E96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2_3_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:141_2_3_2_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_2_3_2_2_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:141_2_3_2_2_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:141_2_3_2_2_1_1"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="polynom1.html#E5:141_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:141_2_3_2_2_1_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_2_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self">Def17</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> = 
0
<b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_2_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="polynom1.html#E2:141_2_3_2_2_1_1"><i><font color="Green">E99</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_2_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="polynom1.html#E2:141_2_3_2_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="polynom1.html#E3:141_2_3_2_2_1_1"><i><font color="Green">E100</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:141_2_3_2_2_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="polynom1.html#E10:141_2_3_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="polynom1.html#E1:141_2_3_2_2_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polynom1.html#E11:141_2_3_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="polynom1.html#E10:141_2_3_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="polynom1.html#E12:141_2_3_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/>
<b>hence </b><a NAME="E16:141_2_3_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E3:141_2_3_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="polynom1.html#E6:141_2_3_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="polynom1.html#E14:141_2_3_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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

<b>thus </b><a NAME="E5:141_2_3"/><i><font color="Green">E88</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>

<b>thus </b><a NAME="E6:141_2_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E7:141_2_3"/>
 <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:141_2_3_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>4</sub></font> ;<br/>
<a NAME="E3:141_2_3_3"/><i><font color="Green">E89</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<a NAME="E4:141_2_3_3"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:141_2_3_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:141_2_3_3"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,0 as  <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></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="141_2_3_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:141_2_3_3_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_2_3_3_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:141_2_3_3_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:141_2_3_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="polynom1.html#E3:141_2_3_3"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:141_2_3_3_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_2_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:141_2_3_3_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:141_2_3_3_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:141_2_3_3_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E4:141_2_3_3"><i><font color="Green">E90</font></i></a>, <a class="txt" href="polynom1.html#E1:141_2_3_3_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self">Def17</a></i>;<br/><b>hence </b><a NAME="E4:141_2_3_3_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:141_2_3_3_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="polynom1.html#E2:141_2_3_3_1_1_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="polynom1.html#K18" target="_self">NatMinor</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E9:141_2_3_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E4:141_2_3_3"><i><font color="Green">E90</font></i></a>, <a class="ref" href="polynom1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E10:141_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E12:141_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:141_2_3_3"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E12:141_2_3_3"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E5:141_2_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T37">FUNCT_7:37</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>
;<br/>
<a NAME="E14:141_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:141_2_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="polynom1.html#E11:141_2_3_3"><i><font color="Green">E91</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:141_2_3_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="polynom1.html#E12:141_2_3_3"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T65">CARD_2:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>
<br/>.= 
<a href="binop_2.html#K48">multnat</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K48">multnat</a>  <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom1.html#E7:141_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="polynom1.html#E11:141_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="polynom1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>
<br/>.= 
<a href="binop_2.html#K48">multnat</a>  <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
<b>by </b><i><a class="txt" href="polynom1.html#E1:141_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="polynom1.html#E4:141_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="setwop_2.html#T4">SETWOP_2:4</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:141"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S2">SETWISEO:sch 2</a>(<a class="txt" href="polynom1.html#E1:141"><i><font color="Green">E77</font></i></a>, <a class="txt" href="polynom1.html#E2:141"><i><font color="Green">E78</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="polynom1.html#V1" target="_self">finite-support</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E4:141"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K19" target="_self">support</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E5:141"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K48">multnat</a>  <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" target="_self">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="polynom1.html#E3:141"><i><font color="Green">E79</font></i></a></i>;<br/>


</div>
