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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>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:35"/><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>
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</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>
 <b>and </b><br/><a NAME="E3:35"/><i><font color="Green">E27</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:35"/><i><font color="Green">E28</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E5:35"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E6:35"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E7:35"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<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></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<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></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E8:35"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</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="E10:35"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E7:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom3.html#E8:35"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xcmplx_1.html#T2">XCMPLX_1:2</a></i>;<br/>
<a NAME="E11:35"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<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></span>)</span> <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#E5:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E12:35"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<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></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E6:35"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:35_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:35_1_1"/><b>then </b>
 <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="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:35_1_1"/><b>then </b>
 <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="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E1:35"><i><font color="Green">E25</font></i></a>, <a class="ref" href="polynom3.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E5:35_1_1"/><b>then </b><i><font color="Green">E36</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="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<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></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E7:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom3.html#E11:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:35_1_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<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></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:35_1_1"/><b>then </b>
 <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> <span class="p2">(<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></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E5:35_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E8:35_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E9:35_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom3.html#E4:35"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:35_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E10:35"><i><font color="Green">E33</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:35_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:35_1_2"/><b>then </b>
 <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#NR2" target="_self">&gt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E5:35_1_2"/><b>then </b>
 <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#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom3.html#E2:35"><i><font color="Green">E26</font></i></a>, <a class="ref" href="polynom3.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E6:35_1_2"/><b>then </b><i><font color="Green">E36</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>2</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> <span class="p2">(<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></span>)</span> <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#E7:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom3.html#E12:35"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:35_1_2"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<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></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E8:35_1_2"/><b>then </b>
 <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<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></span>)</span> <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#E6:35_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E9:35_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E10:35_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom3.html#E3:35"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
