<?xml version="1.0"?>
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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">X</font>, <font color="Maroon">b2</font> be   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span>
 ;<br/>

<a NAME="E2:9"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E3:9"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E4:9"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <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="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:9"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:9"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> = <font color="Maroon">b1</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span>;<br/>
<a NAME="E7:9"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E8:9"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E9:9"/><i><font color="Green">E50</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="E10:9"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:9"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></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="E12:9"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></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="E13:9"/><i><font color="Green">E52</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:9"/><i><font color="Green">E53</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></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"><font color="Maroon">b1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></span>)</span> = 
<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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&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/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">b1</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">X</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/>
<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>;<br/>
<a NAME="E17:9"/><i><font color="Green">E54</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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></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">X</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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E18:9"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E19:9"/><i><font color="Green">E56</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">X</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="E20:9"/>
<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">X</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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></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="E21:9"/><b>then </b><i><font color="Green">E57</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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></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="E22:9"/><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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></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">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E23:9"/><i><font color="Green">E60</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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></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">X</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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">X</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/>
<b>set </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span>;<br/>
<a NAME="E26:9"/><i><font color="Green">E61</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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></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">X</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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E27:9"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E28:9"/><i><font color="Green">E64</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">X</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="E29:9"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E30:9"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E31:9"/><b>then </b>
<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">X</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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></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="E32:9"/><b>then </b><i><font color="Green">E65</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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></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="E33:9"/><i><font color="Green">E66</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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></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">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E34:9"/><i><font color="Green">E67</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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></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">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></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">X</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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">X</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:9_4"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_4_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_4_1_1"><b>suppose </b></a><a NAME="E1:9_4_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow14.html#T3">YELLOW14:3</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="yellow14.html#T3">YELLOW14:3</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_4_1_2"><b>suppose </b></a><a NAME="E1:9_4_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T46">CQC_LANG:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i>, , <a class="ref" href="cqc_lang.html#T46">CQC_LANG:46</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font>

;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E38:9"/><b>then </b>
<font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<a NAME="E39:9"/><b>then </b><i><font color="Green">E71</font></i>: 
<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#K6">-</a> <span class="p1">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">b</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#K3">Sum</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="uproots.html#T17">UPROOTS:17</a></i>;<br/>
<a NAME="E40:9"/><i><font color="Green">E72</font></i>: 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E41:9"/><i><font color="Green">E73</font></i>: 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">b</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/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:9_5"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_5_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_5_1_1"><b>suppose </b></a><a NAME="E1:9_5_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>thus </b><font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="yellow14.html#T3">YELLOW14:3</a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow14.html#T3">YELLOW14:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow14.html#T3">YELLOW14:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</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="9_5_1_2"><b>suppose </b></a><a NAME="E1:9_5_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:9_5_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="cqc_lang.html#T46">CQC_LANG:46</a></i>;<br/><b>thus </b><font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="cqc_lang.html#T46">CQC_LANG:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="cqc_lang.html#T46">CQC_LANG:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</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="E43:9"/><b>then </b>
<font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>hence </b> <a href="uproots.html#K3">Sum</a> <font color="Maroon">b2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">b1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">b</font></span>)</span></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">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="uproots.html#T17">UPROOTS:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">b1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <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>
<b>by </b><i>, <a class="ref" href="glib_004.html#T1" target="_self">Th1</a></i>
;<br/><br/>


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