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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</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 <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#K6" title="PREPOWER:func.6">#R</a> <font color="Maroon" title="c3">c</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">c</font></span>)</span> )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">s</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><span class="kw"> such that </span><br/><a NAME="E2:114"/><span class="lab"><font color="Green" title="E96">A1</font></span>: 
<font color="Maroon" title="c4">s</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E3:114"/><span class="lab"><font color="Green" title="E97">A2</font></span>: 
<font color="Maroon" title="c3">c</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">s</font>
 <span class="kw">and </span><br/><a NAME="E4:114"/>
 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">s</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T67" target="_self" onmouseover="rs('prepower/T67')" onmouseout="rh()">Th67</a></span>;<br/>
<a NAME="E5:114"/><span class="lab"><font color="Green" title="E98">A3</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">s</font></span>)</span> <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">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E3:114"><span class="lab"><font color="Green" title="E97">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="E6:114"/><span class="lab"><font color="Green" title="E99">A4</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">s</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="#E2:114"><span class="lab"><font color="Green" title="E96">A1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:114"/><span class="lab"><font color="Green" title="E100">A5</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#K6" title="PREPOWER:func.6">#R</a> <font color="Maroon" title="c3">c</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">c</font></span>)</span></span><br/>

<a NAME="E8:114"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A6</font></span>: 
<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#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c4">s</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="#E2:114"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="ref" href="prepower.html#T69" target="_self" onmouseover="rs('prepower/T69')" onmouseout="rh()">Th69</a>, <a class="ref" href="prepower.html#T81" target="_self" onmouseover="rs('prepower/T81')" onmouseout="rh()">Th81</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E102">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_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"><span class="p2">(<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#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c4">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> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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"><span class="p2">(<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#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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">#R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<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#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> = 
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</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> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:114"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="txt" href="prepower.html#E107"><span class="lab"><font color="Green" title="E93">Lm11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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>
; <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="E10:114"/>
<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="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="#E7:114"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="ref" href="prepower.html#T81" target="_self" onmouseover="rs('prepower/T81')" onmouseout="rh()">Th81</a></span>;<br/>
<a NAME="E11:114"/><span class="kw">then </span>
<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#K6" title="PREPOWER:func.6">#R</a> <font color="Maroon" title="c3">c</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"><span class="p2">(<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#K5" title="PREPOWER:func.5">#Q</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E3:114"><span class="lab"><font color="Green" title="E97">A2</font></span></a>, <a class="txt" href="#E8:114"><span class="lab"><font color="Green" title="E101">A6</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="E12:114"/>
<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#K6" title="PREPOWER:func.6">#R</a> <font color="Maroon" title="c3">c</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">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:114"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="txt" href="#E8:114"><span class="lab"><font color="Green" title="E101">A6</font></span></a>, <a class="txt" href="#E6:114"><span class="lab"><font color="Green" title="E99">A4</font></span></a>, <a class="txt" href="#E5:114"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E9:114"><span class="lab"><font color="Green" title="E102">A7</font></span></a>, <a class="ref" href="prepower.html#T90" target="_self" onmouseover="rs('prepower/T90')" onmouseout="rh()">Th90</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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