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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:32"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E1:32"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:32_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E4:32_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E3:32"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polynom3.html#E1:32_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:32_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:32_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E8:32_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/><a NAME="E9:32_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E8:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/><div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:32_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:32_1_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E1:32"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E3:32_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E6:32_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:32_1_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E3:32_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:32_1_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E3:32"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polynom3.html#E4:32_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E7:32_1_1_1"/><i><font color="Green">E36</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="polynom3.html#E1:32_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom3.html#E2:32_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom3.html#E4:32_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="polynom3.html#E5:32_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E1:32_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom3.html#E4:32_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E1:32_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom3.html#E2:32_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom3.html#E5:32_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom3.html#E7:32_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E8:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polynom3.html#E1:32_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom3.html#E3:32_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div><a NAME="E11:32_1_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E12:32_1_1"/>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E3:32_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom3.html#E6:32_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom3.html#E9:32_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom3.html#E10:32_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_1_2_1_1_1"/>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E2:32_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_2_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:32_1_2_1_1_2"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:32_1_2_1_1_2"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:32_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E7:32_1_2_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E3:32_1_2_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/><a NAME="E8:32_1_2_1_1_2"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E1:32_1_2_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E9:32_1_2_1_1_2"/><i><font color="Green">E30</font></i>: 
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E7:32_1_2_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/><div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:32_1_2_1_1_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:32_1_2_1_1_2_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E5:32_1_2_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:32_1_2_1_1_2_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E8:32_1_2_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E5:32_1_2_1_1_2_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E1:32_1_2_1_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom3.html#E3:32_1_2_1_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E6:32_1_2_1_1_2_1"/><i><font color="Green">E36</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T20">FINSEQ_5:20</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E1:32_1_2_1_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom3.html#E3:32_1_2_1_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E1:32_1_2_1_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom3.html#E3:32_1_2_1_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="polynom3.html#E5:32_1_2_1_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom3.html#E6:32_1_2_1_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E7:32_1_2_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polynom3.html#E1:32_1_2_1_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom3.html#E2:32_1_2_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div><a NAME="E11:32_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E12:32_1_2_1_1_2"/>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E2:32_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom3.html#E5:32_1_2_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom3.html#E8:32_1_2_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom3.html#E9:32_1_2_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom3.html#E10:32_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:32_1_2"/>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
