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<b>let </b><font color="Maroon" title="c1">r</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> holds <br/><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a></span><br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> implies <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 <b>and </b><br/><a NAME="E3:45"/><i><font color="Green" title="E27">A3</font></i>: 
not <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c3">n1</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> <b> such that </b><br/><a NAME="E5:45"/><i><font color="Green" title="E28">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="seq_1.html#E3:45"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="seq_1.html#T7" target="_self" title="SEQ_1:th.7">Th7</a></i>;<br/>
<a NAME="E6:45"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="seq_1.html#E5:45"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="seq_1.html#T13" target="_self" title="SEQ_1:th.13">Th13</a></i>;<br/>
<a NAME="E7:45"/>
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="seq_1.html#E2:45"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="seq_1.html#T7" target="_self" title="SEQ_1:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E8:45"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_1.html#E1:45"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="seq_1.html#E6:45"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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