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<b>let </b><font color="Maroon" title="c1">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">seq</font> being  <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">N</font>  st <font color="Olive" title="b1">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="b2">seq</font> is <a href="toprns_1.html#V1" title="TOPRNS_1:attr.1">non-zero</a> holds <br/><font color="Olive" title="b1">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Olive" title="b2">seq</font> is <a href="toprns_1.html#V1" title="TOPRNS_1:attr.1">non-zero</a></span><br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">N</font>  st <font color="Maroon" title="c2">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="toprns_1.html#V1" title="TOPRNS_1:attr.1">non-zero</a> holds <br/><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Olive" title="b1">seq</font> is <a href="toprns_1.html#V1" title="TOPRNS_1:attr.1">non-zero</a></span><br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">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="c3">seq</font> is <a href="toprns_1.html#V1" title="TOPRNS_1:attr.1">non-zero</a> implies <font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq</font> is <a href="toprns_1.html#V1" title="TOPRNS_1:attr.1">non-zero</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c2">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="E2:23"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c3">seq</font> is <a href="toprns_1.html#V1" title="TOPRNS_1:attr.1">non-zero</a>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green" title="E19">A3</font></i>: 
not <font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq</font> is <a href="toprns_1.html#V1" title="TOPRNS_1:attr.1">non-zero</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="c4">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="E5:23"/><i><font color="Green" title="E20">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font>
 <b>by </b><i><a class="txt" href="#E3:23"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="toprns_1.html#T4" target="_self" title="TOPRNS_1:th.4">Th4</a></i>;<br/>
<a NAME="E6:23"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font>
 
<b>by </b><i><a class="txt" href="#E5:23"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="toprns_1.html#D3" target="_self" title="TOPRNS_1:def.3">Def3</a></i>;<br/>
<a NAME="E7:23"/>
<font color="Maroon" title="c3">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font>
 
<b>by </b><i><a class="txt" href="#E2:23"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="toprns_1.html#T4" target="_self" title="TOPRNS_1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E8:23"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E6:23"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="euclid.html#T35" title="EUCLID:th.35">EUCLID:35</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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