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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#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="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#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</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#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</font></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:103"/><span class="lab"><font color="Green" title="E84">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> 
 <span class="kw">and </span><br/><a NAME="E2:103"/><span class="lab"><font color="Green" title="E85">A2</font></span>: 
<font color="Maroon" title="c2">b</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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</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#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</font></span>)</span></span><br/>

<a NAME="E3:103"/><span class="lab"><font color="Green" title="E86">A3</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></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="E5:103"/><span class="lab"><font color="Green" title="E87">A4</font></span>: 
<font color="Maroon" title="c4">s1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E6:103"/><span class="lab"><font color="Green" title="E88">A5</font></span>: 
<font color="Maroon" title="c3">c</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="c4">s1</font>
 <span class="kw">and </span><br/><a NAME="E7:103"/>
 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="c4">s1</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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T68" target="_self" title="PREPOWER:th.68">Th80</a></span>;<br/>
<a NAME="E8:103"/><span class="lab"><font color="Green" title="E89">A6</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">s1</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="#E1:103"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E5:103"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="ref" href="prepower.html#T69" target="_self" title="PREPOWER:th.69">Th82</a></span>;<br/>
<a NAME="E9:103"/><span class="lab"><font color="Green" title="E90">A7</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</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="#E2:103"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E5:103"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="ref" href="prepower.html#T69" target="_self" title="PREPOWER:th.69">Th82</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <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> ; <a class="txt" 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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</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> <span class="p1">(<span class="default"><span class="p2">(<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">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="seq_1.html#K1" title="SEQ_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" title="PREPOWER:def.6">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="seq_1.html#K1" title="SEQ_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="#E1:103"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="ref" href="prepower.html#T56" target="_self" title="PREPOWER:th.56">Th67</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#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Q</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="seq_1.html#K1" title="SEQ_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="ref" href="prepower.html#D6" target="_self" title="PREPOWER:def.6">Def7</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#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_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" title="PREPOWER:def.6">Def7</a></span>
; <a class="txt" 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="E11:103"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</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#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></span>;<br/>
<a NAME="E12:103"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</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="#E8:103"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E9:103"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="ref" href="seq_2.html#T14" title="SEQ_2:th.14">SEQ_2:14</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</font> = 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E5:103"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:103"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<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">s1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:103"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E9:103"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="txt" href="#E11:103"><span class="lab"><font color="Green" title="E91">A8</font></span></a>, <a class="ref" href="seq_2.html#T15" title="SEQ_2:th.15">SEQ_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K8" title="PREPOWER:func.8">#Q</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E5:103"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:103"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="txt" href="#E8:103"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K9" title="PREPOWER:func.9">#R</a> <font color="Maroon" title="c3">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E5:103"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:103"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="txt" href="#E9:103"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def8</a></span>
;<br/> <a class="txt" 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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