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<div class="add">

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



<b>assume </b><b>that </b><br/><a NAME="E1:122"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E2:122"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:122"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E1:122"><i><font color="Green">E98</font></i></a>, <a class="ref" href="axioms.html#T20">AXIOMS:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:122"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E1:122"><i><font color="Green">E98</font></i></a>, <a class="txt" href="prepower.html#E4:122"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/>
<i><font color="Green">E102</font></i>: <font color="Maroon">c<sub>1</sub></font> = 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="prepower.html#E4:122"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xcmplx_1.html#T74">XCMPLX_1:74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E2:122"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

;<br/>
<b>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="E9:122"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/>
<a NAME="E10:122"/><i><font color="Green">E104</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:122"><i><font color="Green">E99</font></i></a>, <a class="txt" href="prepower.html#E6:122"><i><font color="Green">E101</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E11:122"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:122"><i><font color="Green">E99</font></i></a>, <a class="txt" href="prepower.html#E9:122"><i><font color="Green">E103</font></i></a>, <a class="ref" href="real_2.html#T200">REAL_2:200</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E12:122"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E13:122"/><i><font color="Green">E107</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="122_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:122_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<i><font color="Green">E109</font></i>: 2 <a href="prepower.html#K3" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 2
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

;<br/>
<a NAME="E3:122_2"/>
2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="prepower.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E4:122_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</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">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:122_2"/>
<span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="prepower.html#E1:122_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E6:122_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="prepower.html#E2:122_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="prepower.html#E4:122_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:122"/>
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#E12:122"><i><font color="Green">E106</font></i></a>, <a class="txt" href="prepower.html#E13:122"><i><font color="Green">E107</font></i></a>);<br/></i>
<a NAME="E15:122"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E16:122"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="prepower.html#E9:122"><i><font color="Green">E103</font></i></a>, <a class="txt" href="prepower.html#E10:122"><i><font color="Green">E104</font></i></a></i>;<br/>
<a NAME="E17:122"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:122"><i><font color="Green">E99</font></i></a>, <a class="txt" href="prepower.html#E15:122"><i><font color="Green">E108</font></i></a>, <a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E18:122"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prepower.html#E7:122"><i><font color="Green">E102</font></i></a>, <a class="txt" href="prepower.html#E11:122"><i><font color="Green">E105</font></i></a>, <a class="txt" href="prepower.html#E17:122"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
