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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K2" target="_self">GeoSeq</a> 
 ;<br/><b>hence </b><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> 0
<b>by </b><i><a class="ref" href="prepower.html#D1" target="_self">Def1</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
;<br/><br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</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> = 
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</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>by </b><i><a class="txt" href="prepower.html#E1:7_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="prepower.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="prepower.html#E1:7_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="prepower.html#D1" target="_self">Def1</a></i>
;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:7"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K2" target="_self">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> 0
<b>by </b><i><a class="txt" href="prepower.html#E2:7"><i><font color="Green">E6</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K2" target="_self">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="prepower.html#D1" target="_self">Def1</a></i>
;<br/>
<a NAME="E5:7"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E6:7"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:7_3"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K2" target="_self">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</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> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="prepower.html#E3:7"><i><font color="Green">E7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="prepower.html#E1:7_3"><i><font color="Green">E10</font></i></a>, <a class="ref" href="prepower.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</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>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K2" target="_self">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</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>by </b><i><a class="ref" href="prepower.html#D1" target="_self">Def1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:7"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="prepower.html#E5:7"><i><font color="Green">E8</font></i></a>, <a class="txt" href="prepower.html#E6:7"><i><font color="Green">E9</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:7"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="prepower.html#K2" target="_self">GeoSeq</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
