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

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<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>let </b><font color="Maroon">b1</font> be   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font>;<br/>







<b>assume </b><a NAME="E1:3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b2</font>
 ;<br/>

<b>set </b><font color="Maroon">B1</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b1</font>;<br/>
<b>set </b><font color="Maroon">B2</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b2</font>;<br/>
<a NAME="E2:3"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E3:3"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</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="E4:3"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <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="E5:3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font></span>)</span> <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#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E6:3"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b1</font></span>)</span> <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/>
<a NAME="E7:3"/><i><font color="Green">E50</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:3"/><i><font color="Green">E51</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b1</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E9:3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font></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">b2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E10:3"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E11:3"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:3"/><b>then </b><i><font color="Green">E52</font></i>: 
<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="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></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">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">B1</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b1</font> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="seq_1.html#D1">SEQ_1:def 1</a>, , <a class="ref" href="polynom1.html#D8">POLYNOM1:def 8</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E15:3"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E16:3"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</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="E17:3"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <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="E18:3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</font></span>)</span> <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#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E19:3"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b2</font></span>)</span> <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/>
<a NAME="E20:3"/><i><font color="Green">E57</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E21:3"/><i><font color="Green">E58</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b2</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></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">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">B2</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">b2</font> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="seq_1.html#D1">SEQ_1:def 1</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="polynom1.html#D8">POLYNOM1:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</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="E25:3"/><i><font color="Green">E60</font></i>: 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f1</font>
 <b>and </b><br/><a NAME="E26:3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D3">UPROOTS:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">F1</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="E28:3"/><i><font color="Green">E63</font></i>: 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F1</font>
 <b>and </b><br/><a NAME="E29:3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">B1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D3">UPROOTS:def 3</a></i>;<br/>
<a NAME="E30:3"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5">UPROOTS:5</a></i>;<br/>
<a NAME="E31:3"/><i><font color="Green">E66</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<a NAME="E32:3"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</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="E33:3"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">B1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5">UPROOTS:5</a></i>;<br/>
<a NAME="E34:3"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<a NAME="E35:3"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">B1</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_3"><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="E1:3_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font>
 ;<br/><a NAME="E2:3_3_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:3_3_1"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/><a NAME="E4:3_3_1"/><b>then </b>
<font color="Maroon">B1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="seq_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/><b>hence </b><a NAME="E5:3_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:3_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">B1</font>
 ;<br/><a NAME="E3:3_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">B1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_3_2"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_3_2_1"><b>suppose </b></a><a NAME="E1:3_3_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">B1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>
;<br/><b>hence </b><a NAME="E3:3_3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_3_2_2"><b>suppose </b></a><a NAME="E1:3_3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:3_3_2_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="seq_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/><b>hence </b><a NAME="E3:3_3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E37:3"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E38:3"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:3_4"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 ;<br/><a NAME="E2:3_4"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>thus </b><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">b1</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">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_004.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">B1</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">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E40:3"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <font color="Maroon">b1</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon">f2</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="E42:3"/><i><font color="Green">E82</font></i>: 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f2</font>
 <b>and </b><br/><a NAME="E43:3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D3">UPROOTS:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">F2</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="E45:3"/><i><font color="Green">E85</font></i>: 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F2</font>
 <b>and </b><br/><a NAME="E46:3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">B2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D3">UPROOTS:def 3</a></i>;<br/>
<a NAME="E47:3"/><i><font color="Green">E87</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5">UPROOTS:5</a></i>;<br/>
<a NAME="E48:3"/><i><font color="Green">E88</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<a NAME="E49:3"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="glib_004.html#E2:3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E50:3"/><i><font color="Green">E90</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5">UPROOTS:5</a></i>;<br/>
<a NAME="E51:3"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<a NAME="E52:3"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">B2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E1:3"><i><font color="Green">E34</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="3_5"><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="E1:3_5_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font>
 ;<br/><a NAME="E2:3_5_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:3_5_1"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/><a NAME="E4:3_5_1"/><b>then </b>
<font color="Maroon">B2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="seq_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/><b>hence </b><a NAME="E5:3_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:3_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">B2</font>
 ;<br/><a NAME="E3:3_5"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">B2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_5_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_5_2"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_5_2_1"><b>suppose </b></a><a NAME="E1:3_5_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">B2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>
;<br/><b>hence </b><a NAME="E3:3_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_5_2_2"><b>suppose </b></a><a NAME="E1:3_5_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:3_5_2_2"/><b>then </b>
<font color="Maroon">B2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="seq_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/><b>hence </b><a NAME="E3:3_5_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E54:3"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:3_6"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font>
 ;<br/><a NAME="E2:3_6"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>thus </b><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">b2</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">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">B2</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">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_004.html#T2" target="_self">Th2</a>, <a class="txt" href="glib_004.html#E2:3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , , <a class="txt" href="glib_004.html#E1:3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E56:3"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <font color="Maroon">b2</font>
 
<b>by </b><i>, , , , <a class="ref" href="glib_004.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_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/><b>assume </b><a NAME="E1:3_7"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:3_7"/><i><font color="Green">E101</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:3_7"/><i><font color="Green">E102</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_7_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_7_1_1"><b>suppose </b></a><a NAME="E1:3_7_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:3_7_1_1"/><i><font color="Green">E104</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:3_7_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">B1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="seq_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/><i><font color="Green">E106</font></i>: <font color="Maroon">B2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>
;<br/><b>thus </b><font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E4:3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_004.html#E3:3"><i><font color="Green">E47</font></i></a>, , , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">B2</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_7_1_2"><b>suppose </b></a><a NAME="E1:3_7_1_2"/><i><font color="Green">E107</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><i><font color="Green">E108</font></i>: <font color="Maroon">B1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">A</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>
;<br/><a NAME="E3:3_7_1_2"/><i><font color="Green">E109</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">b2</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:3_7_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">B2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="seq_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/><b>thus </b><font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E4:3"><i><font color="Green">E48</font></i></a>, , , <a class="txt" href="glib_004.html#E3:3"><i><font color="Green">E47</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">B1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">B2</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E58:3"/><b>then </b>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">B1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E59:3"/>
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">b1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self">Th1</a>, , <a class="ref" href="uproots.html#T17">UPROOTS:17</a></i>;<br/>


</div>
