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<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">s</font> =  <a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> 1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">s</font> =  <a href="seq_1.html#K1" title="SEQ_1:func.1">seq_const</a> 1 as   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a> ;<br/>
<a NAME="E2:90"/>
<font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T57" onmouseover="rs('seq_1/T57')" onmouseout="rh()">SEQ_1:57</a></span>;<br/>
<a NAME="E3:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A1</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:90"/><span class="lab"><font color="Green" title="E76">A2</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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font></span><br/>

<div><span class="lab"><font color="Green" title="E77">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="90_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">nn</font> = <font color="Maroon" title="c4">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</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#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> = 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">nn</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#D6" target="_self" onmouseover="rs('prepower/D6')" onmouseout="rh()">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c1">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="ref" href="prepower.html#T48" target="_self" onmouseover="rs('prepower/T48')" onmouseout="rh()">Th48</a>, <a class="ref" href="seq_1.html#T57" onmouseover="rs('seq_1/T57')" onmouseout="rh()">SEQ_1:57</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:90"/>
<font color="Maroon" title="c1">a</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" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E7:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A4</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c3">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="#E5:90"><span class="lab"><font color="Green" title="E77">A3</font></span></a>, <a class="ref" href="valued_0.html#D18" onmouseover="rs('valued_0/D18')" onmouseout="rh()">VALUED_0:def 18</a></span>;<br/>
<a NAME="E8:90"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E77">A3</font></span></a></span>;<br/>
<a NAME="E9:90"/><span class="kw">then </span>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:90"/>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="txt" href="#E3:90"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="ref" href="prepower.html#D7" target="_self" onmouseover="rs('prepower/D7')" onmouseout="rh()">Def6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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