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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"> for <font color="Olive" title="b1">p</font> being  <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>  st <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>  holds <br/><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Olive" title="b1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  implies <font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:111"/><span class="lab"><font color="Green" title="E78">A1</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</font></span><br/>

<a NAME="E2:111"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">s</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">p</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></span>;<br/>
<span class="lab"><font color="Green" title="E79">A2</font></span>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> = 
<font color="Maroon" title="c3">s</font> <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/>.= 
<font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">s</font> = <font color="Maroon" title="c3">s</font> as   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a> ;<br/>
<div><span class="lab"><font color="Green" title="E80">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">nn</font> = <font color="Maroon" title="c5">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"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">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="c4">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">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/>.= 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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="E7:111"/>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E8:111"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A4</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">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="txt" href="#E6:111"><span class="lab"><font color="Green" title="E80">A3</font></span></a>, <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> <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#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">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/>.= 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111"><span class="lab"><font color="Green" title="E80">A3</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:111"/>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E78">A1</font></span></a>, <a class="txt" href="#E4:111"><span class="lab"><font color="Green" title="E79">A2</font></span></a>, <a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E81">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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