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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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">  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#K5" title="NUMBERS:func.5">0</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</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">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">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">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">s1</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><span class="kw"> such that </span><br/><a NAME="E2:111"/><span class="lab"><font color="Green" title="E94">A1</font></span>: 
<font color="Maroon" title="c4">s1</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E3:111"/><span class="lab"><font color="Green" title="E95">A2</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c4">s1</font>
 <span class="kw">and </span><br/><a NAME="E4:111"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T68" target="_self" onmouseover="rs('prepower/T68')" onmouseout="rh()">Th68</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">s2</font> = <font color="Maroon" title="c3">p</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">s1</font> as   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a> ;<br/>
<span class="kw">assume </span><a NAME="E6:111"/><span class="lab"><font color="Green" title="E96">A3</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">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">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E97">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_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 <br/> ( <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="c4">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<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="c4">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="c4">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<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="c4">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )</span><br/><a NAME="E1:111_1"/><span class="lab"><font color="Green" title="E98">A5</font></span>: 
<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="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="ref" href="prepower.html#T52" target="_self" onmouseover="rs('prepower/T52')" onmouseout="rh()">Th52</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<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="c4">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font>
<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> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="ref" href="prepower.html#T59" target="_self" onmouseover="rs('prepower/T59')" onmouseout="rh()">Th59</a></span>
<br/>.= 
<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="c5">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T9" onmouseover="rs('seq_1/T9')" onmouseout="rh()">SEQ_1:9</a></span>
<br/>.= 
<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="c5">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
<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/><span class="kw">hence </span><a NAME="E3:111_1"/>
( <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="c4">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<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="c4">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="ref" href="prepower.html#D6" target="_self" onmouseover="rs('prepower/D6')" onmouseout="rh()">Def5</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="E8:111"/><span class="lab"><font color="Green" title="E98">A6</font></span>: 
<font color="Maroon" title="c5">s2</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E94">A1</font></span></a></span>;<br/>
<a NAME="E9:111"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A7</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c5">s2</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="ref" href="prepower.html#T69" target="_self" onmouseover="rs('prepower/T69')" onmouseout="rh()">Th69</a></span>;<br/>
<a NAME="E10:111"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c5">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E3:111"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="seq_2.html#T8" onmouseover="rs('seq_2/T8')" onmouseout="rh()">SEQ_2:8</a></span>;<br/>
<a NAME="E11:111"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A8</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">s2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E98">A6</font></span></a>, <a class="txt" href="#E9:111"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="ref" href="prepower.html#D7" target="_self" onmouseover="rs('prepower/D7')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E12:111"/><span class="lab"><font color="Green" title="E101">A9</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c4">s1</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="ref" href="prepower.html#T69" target="_self" onmouseover="rs('prepower/T69')" onmouseout="rh()">Th69</a></span>;<br/>
<a NAME="E13:111"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">s1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E3:111"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="prepower.html#D7" target="_self" onmouseover="rs('prepower/D7')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:111"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">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">#R</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E12:111"><span class="lab"><font color="Green" title="E101">A9</font></span></a>, <a class="txt" href="#E9:111"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="txt" href="#E11:111"><span class="lab"><font color="Green" title="E100">A8</font></span></a>, <a class="txt" href="#E7:111"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="ref" href="prepower.html#T81" target="_self" onmouseover="rs('prepower/T81')" onmouseout="rh()">Th81</a>, <a class="ref" href="prepower.html#T89" target="_self" onmouseover="rs('prepower/T89')" onmouseout="rh()">Th89</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>
