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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">seq</font>, <font color="Olive" title="b2">seq9</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">seq9</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">seq9</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="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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">seq9</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq</font>, <font color="Maroon" title="c3">seq9</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="Red">::  thesis: </font></span></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">seq9</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">seq9</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="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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">seq9</font></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:55"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c2">seq</font> is  <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E2:55"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c3">seq9</font> is  <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">seq9</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="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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">seq9</font></span>)</span></span><br/>

<a NAME="E3:55"/>
 <a href="toprns_1.html#K3" title="TOPRNS_1:func.3">-</a> <font color="Maroon" title="c3">seq9</font> is  <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="toprns_1.html#T45" target="_self" title="TOPRNS_1:th.45">Th45</a></span>;<br/>
<span class="kw">hence </span> <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">seq9</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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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">seq9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="toprns_1.html#T42" target="_self" title="TOPRNS_1:th.42">Th42</a></span>
<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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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">seq9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="toprns_1.html#T46" target="_self" title="TOPRNS_1:th.46">Th46</a></span>
<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="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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">seq9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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