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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c4">R</font> be   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>;<br/>







<a NAME="E1:43"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c4">R</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="ndiff_1.html#D5" target="_self" title="NDIFF_1:def.5">Def5</a></i>;<br/>
<a NAME="E2:43"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c4">R</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#T40" title="VFUNCT_1:th.40">VFUNCT_1:40</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">h</font> be  <a href="ndiff_1.html#V2" title="NDIFF_1:attr.2">convergent_to_0</a> <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>;<br/><a NAME="E1:43_1"/><i><font color="Green" title="E35">A3</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">h</font></span>)</span></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">T</font> )
 <b>by </b><i><a class="ref" href="ndiff_1.html#D5" target="_self" title="NDIFF_1:def.5">Def5</a></i>;<br/><i><font color="Green" title="E36">A4</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">h</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:43"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="ndiff_1.html#T31" target="_self" title="NDIFF_1:th.31">Th31</a></i>
<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ndiff_1.html#T10" target="_self" title="NDIFF_1:th.10">Th10</a></i>
;<br/><b>hence </b><a NAME="E3:43_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">h</font></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:43_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="normsp_1.html#T37" title="NORMSP_1:th.37">NORMSP_1:37</a></i>;<br/><b>thus </b> <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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">T</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:43_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="ndiff_1.html#E2:43_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="normsp_1.html#T45" title="NORMSP_1:th.45">NORMSP_1:45</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43"/>
<font color="Maroon" title="c3">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c4">R</font> is   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:43"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="ndiff_1.html#D5" target="_self" title="NDIFF_1:def.5">Def5</a></i>;<br/>


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