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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>;<br/>





<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E94">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">s1</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><b> such that </b><br/><a NAME="E3:115"/><i><font color="Green" title="E95">A2</font></i>: 
( <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">s1</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> ) )
 <b>by </b><i><a class="ref" href="prepower.html#T80" target="_self" title="PREPOWER:th.80">Th80</a></i>;<br/>
<a NAME="E4:115"/><i><font color="Green" title="E96">A3</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c4">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:115"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="prepower.html#E3:115"><i><font color="Green" title="E95">A2</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self" title="PREPOWER:th.82">Th82</a></i>;<br/>
<a NAME="E5:115"/><b>then </b><i><font color="Green" title="E97">A4</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:115"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="prepower.html#E3:115"><i><font color="Green" title="E95">A2</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self" title="PREPOWER:def.8">Def8</a></i>;<br/>
<a NAME="E6:115"/><b>then </b><i><font color="Green" title="E98">A5</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E1:115"><i><font color="Green" title="E94">A1</font></i></a>, <a class="ref" href="prepower.html#T95" target="_self" title="PREPOWER:th.95">Th95</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:115_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>;<br/><b>hence </b><a NAME="E2:115_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> is   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">s2</font> = <font color="Maroon" title="c3">p</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <font color="Maroon" title="c4">s1</font> as   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a> <b>by </b><i><a class="ref" href="prepower.html#D6" target="_self" title="PREPOWER:def.6">Def6</a></i>;<br/>
<a NAME="E9:115"/><i><font color="Green" title="E99">A6</font></i>: 
<font color="Maroon" title="c5">s2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E3:115"><i><font color="Green" title="E95">A2</font></i></a>, <a class="ref" href="seq_2.html#T21" title="SEQ_2:th.21">SEQ_2:21</a></i>;<br/>
<a NAME="E10:115"/><b>then </b><i><font color="Green" title="E100">A7</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c5">s2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:115"><i><font color="Green" title="E94">A1</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self" title="PREPOWER:th.82">Th82</a></i>;<br/>
<a NAME="E11:115"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c5">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E3:115"><i><font color="Green" title="E95">A2</font></i></a>, <a class="ref" href="seq_2.html#T22" title="SEQ_2:th.22">SEQ_2:22</a></i>;<br/>
<a NAME="E12:115"/><b>then </b><i><font color="Green" title="E101">A8</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c5">s2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:115"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="prepower.html#E9:115"><i><font color="Green" title="E99">A6</font></i></a>, <a class="txt" href="prepower.html#E10:115"><i><font color="Green" title="E100">A7</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self" title="PREPOWER:def.8">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><i><font color="Green" title="E102">A9</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:115"><i><font color="Green" title="E94">A1</font></i></a>, <a class="ref" href="prepower.html#T70" target="_self" title="PREPOWER:th.70">Th70</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font>
<b>by </b><i><a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def7</a></i>
;<br/><a NAME="E2:115_2"/>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E1:115"><i><font color="Green" title="E94">A1</font></i></a>, <a class="ref" href="prepower.html#T63" target="_self" title="PREPOWER:th.63">Th63</a></i>;<br/><b>hence </b><a NAME="E3:115_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="txt" href="prepower.html#E1:115_2"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:115"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E4:115"><i><font color="Green" title="E96">A3</font></i></a>, <a class="txt" href="prepower.html#E5:115"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="prepower.html#E6:115"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="prepower.html#E10:115"><i><font color="Green" title="E100">A7</font></i></a>, <a class="txt" href="prepower.html#E12:115"><i><font color="Green" title="E101">A8</font></i></a>, <a class="ref" href="prepower.html#T103" target="_self" title="PREPOWER:th.103">Th103</a></i>;<br/>


</div>
