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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">seq</font>, <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">seq</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a> &amp; <font color="Olive" title="b2">seq'</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a> holds <br/> <a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq</font> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <font color="Olive" title="b2">seq'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Olive" title="b1">seq</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Olive" title="b2">seq'</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">seq</font>, <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">seq</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a> &amp; <font color="Maroon" title="c3">seq'</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a> implies  <a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <font color="Maroon" title="c3">seq'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">seq'</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <font color="Maroon" title="c3">seq'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span><br/>

<a NAME="E3:50"/>
 <a href="toprns_1.html#K3" title="TOPRNS_1:func.3">-</a> <font color="Maroon" title="c3">seq'</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E2:50"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="toprns_1.html#T45" target="_self" title="TOPRNS_1:th.45">Th45</a></i>;<br/>
<b>hence </b> <a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="toprns_1.html#K4" title="TOPRNS_1:func.4">-</a> <font color="Maroon" title="c3">seq'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <span class="p2">(<span class="default"><a href="toprns_1.html#K3" title="TOPRNS_1:func.3">-</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprns_1.html#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="toprns_1.html#T42" target="_self" title="TOPRNS_1:th.42">Th42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p2">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprns_1.html#E2:50"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="toprns_1.html#T46" target="_self" title="TOPRNS_1:th.46">Th46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></i>
;<br/> <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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