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<div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</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>assume </b><a NAME="E1:97"/><i><font color="Green" title="E78">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">sb</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><b> such that </b><br/><a NAME="E3:97"/><i><font color="Green" title="E79">A2</font></i>: 
( <font color="Maroon" title="c4">sb</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">sb</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">sb</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) )
 <b>by </b><i><a class="ref" href="prepower.html#T79" target="_self" title="PREPOWER:th.79">Th79</a></i>;<br/>
<a NAME="E4:97"/><i><font color="Green" title="E80">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">sb</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#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="prepower.html#E3:97"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self" title="PREPOWER:th.82">Th82</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">sc</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><b> such that </b><br/><a NAME="E6:97"/><i><font color="Green" title="E81">A4</font></i>: 
( <font color="Maroon" title="c5">sc</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">c</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="c5">sc</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="c5">sc</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font> ) )
 <b>by </b><i><a class="ref" href="prepower.html#T79" target="_self" title="PREPOWER:th.79">Th79</a></i>;<br/>
<a NAME="E7:97"/><i><font color="Green" title="E82">A5</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">sc</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#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="prepower.html#E6:97"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self" title="PREPOWER:th.82">Th82</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><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/><a NAME="E1:97_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">sb</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">sc</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">sb</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c5">sc</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/><b>hence </b><a NAME="E2:97_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">sb</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c5">sc</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">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="c6">s</font> = <font color="Maroon" title="c4">sb</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c5">sc</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="E10:97"/><i><font color="Green" title="E83">A6</font></i>: 
<font color="Maroon" title="c6">s</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#E3:97"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="prepower.html#E6:97"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/>
<a NAME="E11:97"/><i><font color="Green" title="E84">A7</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E3:97"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="prepower.html#E6:97"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>;<br/>
<a NAME="E12:97"/><i><font color="Green" title="E85">A8</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <font color="Maroon" title="c6">s</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#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="prepower.html#E10:97"><i><font color="Green" title="E83">A6</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self" title="PREPOWER:th.82">Th82</a></i>;<br/>
<div><i><font color="Green" title="E86">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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/><b>thus </b><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="c6">s</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font> = 
<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="c6">s</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">sb</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">sc</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<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">sb</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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="c5">sc</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="prepower.html#T64" target="_self" title="PREPOWER:th.64">Th64</a></i>
<br/>.= 
<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">sb</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <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="c5">sc</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def7</a></i>
<br/>.= 
<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">sb</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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="c5">sc</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def7</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><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#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font></span>)</span> = 
 <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="c6">s</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="prepower.html#E10:97"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="prepower.html#E11:97"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="prepower.html#E12:97"><i><font color="Green" title="E85">A8</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self" title="PREPOWER:def.8">Def8</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <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">sb</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <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="c5">sc</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E13:97"><i><font color="Green" title="E86">A9</font></i></a>, <a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <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">sb</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <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="c5">sc</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E4:97"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="prepower.html#E7:97"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
<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="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <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="c5">sc</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="prepower.html#E3:97"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="prepower.html#E4:97"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self" title="PREPOWER:def.8">Def8</a></i>
<br/>.= 
<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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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="c3">c</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="prepower.html#E6:97"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="prepower.html#E7:97"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self" title="PREPOWER:def.8">Def8</a></i>
;<br/>


</div>
