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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</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">seq1</font>, <font color="Olive" title="b2">seq2</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">seq1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> &amp; <font color="Olive" title="b2">seq2</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Olive" title="b2">seq2</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> <font color="Maroon" title="c1">X</font> holds <br/> <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b2">seq2</font></span><br/><b>let </b><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</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">seq1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> &amp; <font color="Maroon" title="c3">seq2</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</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> <font color="Maroon" title="c1">X</font> implies  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c3">seq2</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green" title="E7">A3</font></i>: 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</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> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font></span><br/>

<a NAME="E4:4"/>
<span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</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> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E2:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="#E3:4"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="normsp_1.html#T43" title="NORMSP_1:th.43">NORMSP_1:43</a></i>;<br/>
<a NAME="E5:4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font>
 
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>;<br/>
<a NAME="E6:4"/><b>then </b>
<span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font>
 
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>;<br/>
<a NAME="E7:4"/><b>then </b>
<span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>;<br/>
<b>hence </b><a NAME="E8:4"/>
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c3">seq2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>; <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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