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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c3">seq</font> is  <a href="toprns_1.html#V1" title="TOPRNS_1:attr.1">non-zero</a> 
 <span class="kw">and </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E5:26"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="toprns_1.html#T3" target="_self" title="TOPRNS_1:th.3">Th4</a></span>;<br/>
<a NAME="E6:26"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="toprns_1.html#T3" target="_self" title="TOPRNS_1:th.3">Th4</a></span>;<br/>
<a NAME="E7:26"/>
<font color="Maroon" title="c2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="toprns_1.html#T5" target="_self" title="TOPRNS_1:th.5">LmDef3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:26"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</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/>


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