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

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E1:23"/><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> <font color="Maroon">S</font>
 ;<br/>

<b>set </b><font color="Maroon">cS</font> =  <a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">f</font> = <font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span>;<br/>
<a NAME="E2:23"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:23"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:23"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:23"/><i><font color="Green">E45</font></i>: 
 <a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:23"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:23"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E9:23"/><i><font color="Green">E50</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#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#D1">SEQ_1:def 1</a></i>;<br/>
<a NAME="E10:23"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E11:23"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E13:23"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">cs</font> =  <a href="uproots.html#K1" target="_self">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>;<br/>
<a NAME="E14:23"/><i><font color="Green">E51</font></i>: 
 <a href="uproots.html#K1" target="_self">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E16:23"/><i><font color="Green">E52</font></i>: 
 <a href="uproots.html#K3" target="_self">Sum</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E17:23"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E18:23"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E19:23"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:23"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:23"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E22:23"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E23:23"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E24:23"/><i><font color="Green">E58</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">S</font> or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">S</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>suppose </b></a><a NAME="E1:23_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:23_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E4:23_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T1">FSM_1:1</a></i>;<br/><b>set </b><font color="Maroon">h</font> = <font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0;<br/><a NAME="E5:23_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><a NAME="E6:23_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">gr</font> = <font color="Maroon">g</font>, <font color="Maroon">fr</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><b>set </b><font color="Maroon">F</font> = <font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <font color="Maroon">S</font>
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>
;<br/><a NAME="E9:23_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">dd</font> = <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> ;<br/><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:23_1_1_2"/><i><font color="Green">E65</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> ) )
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:23_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span>
 <b>and </b><br/><a NAME="E5:23_1_1_2"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</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/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E7:23_1_1_2"/>
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E8:23_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</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="E9:23_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E10:23_1_1_2"/>
not <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><a NAME="E11:23_1_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:23_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span> 
 ;<br/>

<a NAME="E2:23_1_1_3"/><b>then </b>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 0 )
 
;<br/>
<b>hence </b><a NAME="E3:23_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">dd</font> = <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="uproots.html#T2" target="_self">Th2</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E13:23_1_1"/>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">dd</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">dd</font> &amp; <font color="Maroon">dd</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">cdd</font> =  <a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">dd</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon">cdi</font> = <font color="Maroon">cdd</font> <a href="funct_1.html#K2">"</a> ;<br/><b>reconsider </b><font color="Maroon">cdi</font> = <font color="Maroon">cdd</font> <a href="funct_1.html#K2">"</a>  as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">dd</font>, <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">dd</font></span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">cadd</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">dd</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="graph_5.html#T5">GRAPH_5:5</a></i>;<br/><a NAME="E17:23_1_1"/><i><font color="Green">E70</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">cadd</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">dd</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">cdi</font> = <font color="Maroon">cdi</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">dd</font>, <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/><b>set </b><font color="Maroon">cSr</font> = <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span>;<br/><a NAME="E19:23_1_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/><a NAME="E20:23_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E21:23_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_1_1_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span></span>)</span>
 ;<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="E3:23_1_1_4_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:23_1_1_4_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span></span>)</span> <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="E5:23_1_1_4_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font> )
 <b>by </b><i><a class="txt" href="uproots.html#E3:23"><i><font color="Green">E43</font></i></a>, <a class="ref" href="relat_1.html#T86">RELAT_1:86</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E7:23_1_1_4_1"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E8:23_1_1_4_1"/>
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="txt" href="uproots.html#E5:23"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E9:23_1_1_4_1"/><b>then </b>
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E5:23"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E10:23_1_1_4_1"/><b>then </b>
<font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E5:23"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E11:23_1_1_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uproots.html#E6:23"><i><font color="Green">E46</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:23_1_1_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</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="E4:23_1_1_4"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span>
 <b>and </b><br/><a NAME="E5:23_1_1_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dd</font>
 ;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:23_1_1_4_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:23_1_1_4_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:23_1_1_4_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E6:23_1_1_4_2"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E3:23"><i><font color="Green">E43</font></i></a>, <a class="txt" href="uproots.html#E6:23"><i><font color="Green">E46</font></i></a>, <a class="txt" href="uproots.html#E5:23"><i><font color="Green">E45</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><a NAME="E7:23_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23"><i><font color="Green">E44</font></i></a>, , , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uproots.html#E5:23"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/></div><b>end;</b></div><a NAME="E23:23_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E24:23_1_1"/><b>then </b>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i>, <a class="txt" href="uproots.html#E1:23"><i><font color="Green">E42</font></i></a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/><a NAME="E25:23_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T81">CARD_1:81</a></i>;<br/><a NAME="E26:23_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">dd</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>;<br/><a NAME="E27:23_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">dd</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">dd</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E28:23_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">dd</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E29:23_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">dd</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/><a NAME="E30:23_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">cdd</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">dd</font>
 <b>by </b><i/>;<br/><a NAME="E31:23_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">cdd</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E32:23_1_1"/><i><font color="Green">E111</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">cdd</font> <a href="hidden.html#R1">=</a> <font color="Maroon">dd</font>
 <b>by </b><i/>;<br/><a NAME="E33:23_1_1"/><i><font color="Green">E112</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">cdi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E34:23_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">cdd</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/><a NAME="E35:23_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">cdi</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/><a NAME="E36:23_1_1"/><i><font color="Green">E115</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">cdi</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="txt" href="uproots.html#E5:23"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">z</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E38:23_1_1"/><i><font color="Green">E116</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">dd</font>
 <b>and </b><br/><a NAME="E39:23_1_1"/><i><font color="Green">E117</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">dd</font> holds <br/><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E40:23_1_1"/><i><font color="Green">E118</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:23_1_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">z</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="E3:23_1_1_5"/><i><font color="Green">E119</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">z</font>
 <b>and </b><br/><a NAME="E4:23_1_1_5"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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="E5:23_1_1_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E19:23"><i><font color="Green">E54</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E16:23"><i><font color="Green">E52</font></i></a>, <a class="txt" href="uproots.html#E20:23"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E6:23_1_1_5"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E19:23"><i><font color="Green">E54</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E7:23_1_1_5"/>
<font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E9:23"><i><font color="Green">E50</font></i></a>, <a class="txt" href="uproots.html#E19:23"><i><font color="Green">E54</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E8:23_1_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i><a class="txt" href="uproots.html#E23:23"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:23_1_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E10:23_1_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="uproots.html#E22:23"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">z</font>;<br/><a NAME="E41:23_1_1"/><i><font color="Green">E123</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_6"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<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="E3:23_1_1_6"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:23_1_1_6"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="uproots.html#E22:23"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:23_1_1_6"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E7:23_1_1_6"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E8:23_1_1_6"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E9:23_1_1_6"/><i><font color="Green">E135</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E10:23_1_1_6"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E23:23"><i><font color="Green">E57</font></i></a>, <a class="txt" href="uproots.html#E24:23"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:23_1_1_6"/><b>then </b>
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E12:23_1_1_6"/><b>then </b>
not <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><b>hence </b><a NAME="E13:23_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E24:23"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uproots.html#E23:23"><i><font color="Green">E57</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_1_1_7_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_7_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> )
 </span><b>by </b><i><a class="txt" href="uproots.html#E22:23"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_7_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_1_7_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_7_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_1_7_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:23_1_1_7"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_7_2"/>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_2_1"><b>suppose </b></a><a NAME="E1:23_1_1_7_2_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_1_7_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E22:23"><i><font color="Green">E56</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:23_1_1_7_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_2_2"><b>suppose </b></a><a NAME="E1:23_1_1_7_2_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_1_7_2_2"/>
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="uproots.html#E22:23"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E3:23_1_1_7_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E23:23"><i><font color="Green">E57</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:23_1_1_7_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E5:23_1_1_7_2_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="uproots.html#E22:23"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>hence </b><a NAME="E6:23_1_1_7_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E44:23_1_1"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E45:23_1_1"/><b>then </b><i><font color="Green">E140</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_8"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_8_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> )
 </span><b>by </b><i><a class="txt" href="uproots.html#E24:23"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uproots.html#E22:23"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_8_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_8_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:23_1_1_8_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E20:23"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:23_1_1_8_1_1"/><b>then </b><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E4:23_1_1_8_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:23_1_1_8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/><a NAME="E6:23_1_1_8_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E7:23_1_1_8_1_1"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_8_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_8_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_8_1_2"/><i><font color="Green">E145</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">z</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_8_1_2"/><b>then </b><i><font color="Green">E174</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E3:23_1_1_8_1_2"/>
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_8_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a>, , <a class="txt" href="uproots.html#E17:23"><i><font color="Green">E53</font></i></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">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">z</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>, <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="uproots.html#E23:23"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/><a NAME="E48:23_1_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</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><b>that </b><br/><a NAME="E1:23_1_1_9"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:23_1_1_9"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E3:23_1_1_9"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_9_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> ) )
 </span><b>by </b><i><a class="txt" href="uproots.html#E19:23"><i><font color="Green">E54</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_9_1_1"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:23_1_1_9_1_1"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E20:23"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green">E181</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E4:23_1_1_9_1_1"/>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="uproots.html#E20:23"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green">E182</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E6:23_1_1_9_1_1"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E7:23_1_1_9_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E8:23_1_1_9_1_1"/><b>then </b><i><font color="Green">E183</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/><a NAME="E9:23_1_1_9_1_1"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E10:23_1_1_9_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E11:23_1_1_9_1_1"/><b>then </b><i><font color="Green">E184</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/><a NAME="E12:23_1_1_9_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E13:23_1_1_9_1_1"/><b>then </b><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/><a NAME="E14:23_1_1_9_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>hence </b><a NAME="E15:23_1_1_9_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_9_1_2"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E187</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> = 
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E16:23"><i><font color="Green">E52</font></i></a></i>
;<br/><a NAME="E3:23_1_1_9_1_2"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="uproots.html#E20:23"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green">E188</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:23_1_1_9_1_2"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E6:23_1_1_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:23_1_1_9_1_2"/><b>then </b><i><font color="Green">E189</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon">ccc</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E9:23_1_1_9_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green">E64</font></i></a>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:23_1_1_9_1_2"/><b>then </b>
<font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:23_1_1_9_1_2"/><b>then </b>
<font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E12:23_1_1_9_1_2"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:23_1_1_9_1_2"/>
<font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">cdi</font>
 <b>by </b><i>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E14:23_1_1_9_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="uproots.html#E9:23"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1_3"><b>suppose </b></a><a NAME="E1:23_1_1_9_1_3"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E193</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E16:23"><i><font color="Green">E52</font></i></a></i>
;<br/><a NAME="E3:23_1_1_9_1_3"/>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="uproots.html#E20:23"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green">E194</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:23_1_1_9_1_3"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E6:23_1_1_9_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:23_1_1_9_1_3"/><b>then </b><i><font color="Green">E195</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon">ccc</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E9:23_1_1_9_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green">E64</font></i></a>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:23_1_1_9_1_3"/><b>then </b>
<font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:23_1_1_9_1_3"/><b>then </b>
<font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E12:23_1_1_9_1_3"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:23_1_1_9_1_3"/>
<font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">cdi</font>
 <b>by </b><i>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E14:23_1_1_9_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="uproots.html#E9:23"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1_4"><b>suppose </b></a><a NAME="E1:23_1_1_9_1_4"/><i><font color="Green">E197</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E198</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E16:23"><i><font color="Green">E52</font></i></a></i>
;<br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> = 
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E16:23"><i><font color="Green">E52</font></i></a></i>
;<br/><a NAME="E4:23_1_1_9_1_4"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E5:23_1_1_9_1_4"/>
( <font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E6:23_1_1_9_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_1_1_10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_10_1"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:23_1_1_10_1"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_10_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> )
 </span><b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_4.html#T13">FUNCT_4:13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_10_1_1_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:23_1_1_10_1_1_1"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E20:23"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green">E203</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E4:23_1_1_10_1_1_1"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E5:23_1_1_10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E6:23_1_1_10_1_1_1"/><b>then </b><i><font color="Green">E204</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><a NAME="E7:23_1_1_10_1_1_1"/>
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_10_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, , <a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_10_1_1_2"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_10_1_1_2"/><b>then </b><i><font color="Green">E206</font></i>: 
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:23_1_1_10_1_1_2"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_10_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a>, , <a class="txt" href="uproots.html#E17:23"><i><font color="Green">E53</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:23_1_1_10"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_10_2"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">gr</font> or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">gr</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">n</font> ) )
 </span><b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_2_1"><b>suppose </b></a><a NAME="E1:23_1_1_10_2_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">gr</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_10_2_1"/><b>then </b><i><font color="Green">E209</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:23_1_1_10_2_1"/><b>then </b><i><font color="Green">E210</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_10_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_10_2_1_1"/><i><font color="Green">E211</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 ;<br/><i><font color="Green">E212</font></i>: <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> = 
<font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , , <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/><a NAME="E3:23_1_1_10_2_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_10_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><b>then </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p5">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , , <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
<br/>.= 
<font color="Maroon">j</font>
<b>by </b><i>, , <a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/><b>hence </b><a NAME="E6:23_1_1_10_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="uproots.html#E23:23"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_2_2"><b>suppose </b></a><a NAME="E1:23_1_1_10_2_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">gr</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:23_1_1_10_2_2"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span>
 <b>and </b><br/><a NAME="E4:23_1_1_10_2_2"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">gr</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E5:23_1_1_10_2_2"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">cdd</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="uproots.html#E5:23"><i><font color="Green">E45</font></i></a>, <a class="txt" href="uproots.html#E6:23"><i><font color="Green">E46</font></i></a>, <a class="ref" href="uproots.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">cdd</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">cdd</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">cdd</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="uproots.html#E16:23"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">cdd</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="uproots.html#E5:23"><i><font color="Green">E45</font></i></a>, <a class="ref" href="uproots.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
<font color="Maroon">j</font>
<b>by </b><i>, <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:23_1_1_10_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , <a class="txt" href="uproots.html#E23:23"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E50:23_1_1"/><b>then </b><i><font color="Green">E216</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E51:23_1_1"/><b>then </b>
<font color="Maroon">p</font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><a NAME="E52:23_1_1"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E24:23"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/><a NAME="E53:23_1_1"/><i><font color="Green">E218</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E23:23"><i><font color="Green">E57</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_11"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_11_1"/>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_11_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_11_1_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:23_1_1_11_1_1"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a></i>;<br/><b>then </b><i><font color="Green">E222</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="uproots.html#E16:23"><i><font color="Green">E52</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a></i>
;<br/><a NAME="E5:23_1_1_11_1_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="uproots.html#E23:23"><i><font color="Green">E57</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">px</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E7:23_1_1_11_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">cdi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:23_1_1_11_1_1"/><b>then </b><i><font color="Green">E224</font></i>: 
<font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E9:23_1_1_11_1_1"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">cdi</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">dd</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>reconsider </b><font color="Maroon">cdix</font> = <font color="Maroon">cdi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>thus </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">cdix</font>
<b>by </b><i><a class="txt" href="uproots.html#E4:23"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, , , <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">px</font>
<b>by </b><i>, <a class="ref" href="uproots.html#T1" target="_self">Th1</a>, <a class="txt" href="uproots.html#E4:23"><i><font color="Green">E44</font></i></a>, , , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_11_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_11_1_2"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E228</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_11_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_11_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dd</font>
 ;<br/><a NAME="E2:23_1_1_11_1_2_1"/><b>then </b>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><b>hence </b><a NAME="E3:23_1_1_11_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:23_1_1_11_1_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E4:23_1_1_11_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E5:23_1_1_11_1_2"/><b>then </b><i><font color="Green">E229</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:23_1_1_11_1_2"/><b>then </b><i><font color="Green">E230</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/><i><font color="Green">E231</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p4">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="uproots.html#E14:23"><i><font color="Green">E51</font></i></a>, <a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>then reconsider </b><font color="Maroon">px</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><b>thus </b><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">px</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">px</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">px</font>
<b>by </b><i>, , , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E55:23_1_1"/><b>then </b><i><font color="Green">E232</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><i><font color="Green">E233</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span> = 
<a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D10">RVSUM_1:def 10</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">fr</font>
<b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="finsop_1.html#T8">FINSOP_1:8</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#D10">RVSUM_1:def 10</a></i>
;<br/><a NAME="E57:23_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T111">RVSUM_1:111</a></i>;<br/><b>then </b> <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">gr</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">gr</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>

<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">gr</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="finsop_1.html#K1">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
;<br/><b>hence </b><a NAME="E59:23_1_1"/>
 <a href="uproots.html#K3" target="_self">Sum</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green">E63</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>suppose </b></a><a NAME="E1:23_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2"/>
 <a href="uproots.html#K3" target="_self">Sum</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , <a class="ref" href="triang_1.html#T3">TRIANG_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
