<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">seq1</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">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> ) iff <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> )</span><br/>



<b>thus </b><a NAME="E1:36"/>
( <font color="Maroon" title="c1">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> implies <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</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"> ( <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> implies ( <font color="Maroon" title="c1">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:36_1"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 <b>and </b><br/><a NAME="E2:36_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c2">seq1</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"> <font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</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><br/><a NAME="E1:36_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</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#E1:36_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="seq_1.html#T7" target="_self" title="SEQ_1:th.7">Th7</a></i>;<br/><a NAME="E2:36_1_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</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#E2:36_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="seq_1.html#T7" target="_self" title="SEQ_1:th.7">Th7</a></i>;<br/><a NAME="E3:36_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">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">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T12" target="_self" title="SEQ_1:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</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#E1:36_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="seq_1.html#E2:36_1_1"><i><font color="Green" title="E24">A4</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:36_1"/>
<font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T7" target="_self" title="SEQ_1:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:36"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</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"> ( <font color="Maroon" title="c1">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> &amp; <font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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"> <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</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><br/><a NAME="E1:36_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">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">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T12" target="_self" title="SEQ_1:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E2:36_2"/>
<font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</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#E2:36"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="seq_1.html#T7" target="_self" title="SEQ_1:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:36"/>
<font color="Maroon" title="c1">seq</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T7" target="_self" title="SEQ_1:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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"> <font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</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><br/><a NAME="E1:36_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">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">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T12" target="_self" title="SEQ_1:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E2:36_3"/>
<font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</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#E2:36"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="seq_1.html#T7" target="_self" title="SEQ_1:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:36"/>
<font color="Maroon" title="c2">seq1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T7" target="_self" title="SEQ_1:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
