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<span class="kw">let </span><font color="Maroon" title="c1">a</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 1 <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">s</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><span class="kw"> such that </span><br/><a NAME="E2:110"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
<font color="Maroon" title="c2">s</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E3:110"/><span class="lab"><font color="Green" title="E78">A2</font></span>: 
<font color="Maroon" title="c1">a</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="c2">s</font>
 <span class="kw">and </span><br/><a NAME="E4:110"/>
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="c2">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T79" target="_self" title="PREPOWER:th.79">Th79</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E79">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">1 <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">nn</font> = <font color="Maroon" title="c3">n</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default">1 <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> = 
1 <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">nn</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def7</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T62" target="_self" title="PREPOWER:th.62">Th62</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:110"/><span class="kw">then </span>
1 <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c2">s</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D18" title="VALUED_0:def.18">VALUED_0:def 18</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E80">A4</font></span>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default">1 <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c2">s</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110"><span class="lab"><font color="Green" title="E79">A3</font></span></a></span>
;<br/>
<a NAME="E8:110"/>
1 <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c2">s</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="prepower.html#T82" target="_self" title="PREPOWER:th.82">Th82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:110"/>
1 <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E7:110"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="ref" href="prepower.html#D8" target="_self" title="PREPOWER:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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