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<span class="kw">let </span><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a> 
 <span class="kw">and </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 not <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</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"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:32"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c2">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="seq_1.html#T5" target="_self" onmouseover="rs('seq_1/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E5:32"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c2">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" onmouseover="rs('valued_1/T10')" onmouseout="rh()">VALUED_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:32"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="seq_1.html#T5" target="_self" onmouseover="rs('seq_1/T5')" onmouseout="rh()">Th5</a>, <a class="ref" href="xcmplx_1.html#T202" onmouseover="rs('xcmplx_1/T202')" onmouseout="rh()">XCMPLX_1:202</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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