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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">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>assume </b><a NAME="E1:107"/><i><font color="Green" title="E87">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">b</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">p</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a><b> such that </b><br/><a NAME="E3:107"/><i><font color="Green" title="E88">A2</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">c</font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T22" title="RAT_1:th.22">RAT_1:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">q</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a><b> such that </b><br/><a NAME="E5:107"/><i><font color="Green" title="E89">A3</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c5">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="prepower.html#E3:107"><i><font color="Green" title="E88">A2</font></i></a>, <a class="ref" href="rat_1.html#T22" title="RAT_1:th.22">RAT_1:22</a></i>;<br/>
<a NAME="E6:107"/><i><font color="Green" title="E90">A4</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:107"><i><font color="Green" title="E87">A1</font></i></a>, <a class="txt" href="prepower.html#E5:107"><i><font color="Green" title="E89">A3</font></i></a>, <a class="ref" href="prepower.html#T75" target="_self" title="PREPOWER:th.75">Th75</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">s1</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><b> such that </b><br/><a NAME="E8:107"/><i><font color="Green" title="E91">A5</font></i>: 
( <font color="Maroon" title="c6">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">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="c6">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="c6">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">c</font> ) )
 <b>by </b><i><a class="ref" href="prepower.html#T80" target="_self" title="PREPOWER:th.80">Th80</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">s2</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><b> such that </b><br/><a NAME="E10:107"/><i><font color="Green" title="E92">A6</font></i>: 
( <font color="Maroon" title="c7">s2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">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="c7">s2</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="c7">s2</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">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="E11:107"/><i><font color="Green" title="E93">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="c6">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#E1:107"><i><font color="Green" title="E87">A1</font></i></a>, <a class="txt" href="prepower.html#E8:107"><i><font color="Green" title="E91">A5</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self" title="PREPOWER:th.82">Th82</a></i>;<br/>
<a NAME="E12:107"/><i><font color="Green" title="E94">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="c7">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#E1:107"><i><font color="Green" title="E87">A1</font></i></a>, <a class="txt" href="prepower.html#E10:107"><i><font color="Green" title="E92">A6</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="107_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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="E1:107_1"/>
<font color="Maroon" title="c6">s1</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">c</font>
 <b>by </b><i><a class="txt" href="prepower.html#E8:107"><i><font color="Green" title="E91">A5</font></i></a></i>;<br/><a NAME="E2:107_1"/><b>then </b>
<font color="Maroon" title="c6">s1</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="prepower.html#E3:107"><i><font color="Green" title="E88">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:107_1"/><b>then </b>
<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">s1</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="prepower.html#E1:107"><i><font color="Green" title="E87">A1</font></i></a>, <a class="ref" href="prepower.html#T74" target="_self" title="PREPOWER:th.74">Th74</a></i>;<br/><b>hence </b><a NAME="E4:107_1"/>
<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">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 <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>
<a NAME="E14:107"/><b>then </b>
 <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">s1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E11:107"><i><font color="Green" title="E93">A7</font></i></a>, <a class="ref" href="prepower.html#T2" target="_self" title="PREPOWER:th.2">Th2</a></i>;<br/>
<a NAME="E15:107"/><b>then </b><i><font color="Green" title="E95">A9</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">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:107"><i><font color="Green" title="E87">A1</font></i></a>, <a class="txt" href="prepower.html#E8:107"><i><font color="Green" title="E91">A5</font></i></a>, <a class="txt" href="prepower.html#E11:107"><i><font color="Green" title="E93">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="107_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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="E1:107_2"/>
<font color="Maroon" title="c7">s2</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="prepower.html#E10:107"><i><font color="Green" title="E92">A6</font></i></a></i>;<br/><a NAME="E2:107_2"/><b>then </b>
<font color="Maroon" title="c7">s2</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="prepower.html#E5:107"><i><font color="Green" title="E89">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:107_2"/><b>then </b>
<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="c7">s2</font> <a href="prepower.html#K8" title="PREPOWER:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="prepower.html#E1:107"><i><font color="Green" title="E87">A1</font></i></a>, <a class="ref" href="prepower.html#T74" target="_self" title="PREPOWER:th.74">Th74</a></i>;<br/><b>hence </b><a NAME="E4:107_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="c7">s2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font>
 <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>
<a NAME="E17:107"/><b>then </b>
 <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="c7">s2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E12:107"><i><font color="Green" title="E94">A8</font></i></a>, <a class="ref" href="prepower.html#T3" target="_self" title="PREPOWER:th.3">Th3</a></i>;<br/>
<a NAME="E18:107"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:107"><i><font color="Green" title="E87">A1</font></i></a>, <a class="txt" href="prepower.html#E10:107"><i><font color="Green" title="E92">A6</font></i></a>, <a class="txt" href="prepower.html#E12:107"><i><font color="Green" title="E94">A8</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self" title="PREPOWER:def.8">Def8</a></i>;<br/>
<a NAME="E19:107"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E6:107"><i><font color="Green" title="E90">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E20:107"/>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K10" title="PREPOWER:func.10">#R</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="prepower.html#E15:107"><i><font color="Green" title="E95">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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