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<b>let </b><font color="Maroon">T</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">R1</font> be   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">R2</font> be   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>







<a NAME="E1:42"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">R1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">R2</font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:42"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">R1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R2</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#T38">VFUNCT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be  <a href="ndiff_1.html#V2" target="_self">convergent_to_0</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/><a NAME="E1:42_1"/><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/><a NAME="E2:42_1"/><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R2</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/><i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R2</font></span>)</span> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">R2</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R2</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E4:42_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R2</font></span>)</span> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i>, , <a class="ref" href="normsp_1.html#T34">NORMSP_1:34</a></i>;<br/><b>thus </b> <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R2</font></span>)</span> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="normsp_1.html#T42">NORMSP_1:42</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:42"/>
<font color="Maroon">R1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R2</font> is   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/>

<a NAME="E5:42"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">R1</font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">R2</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i>, <a class="ref" href="vfunct_1.html#T38">VFUNCT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be  <a href="ndiff_1.html#V2" target="_self">convergent_to_0</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/><a NAME="E1:42_2"/><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/><a NAME="E2:42_2"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R2</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/><i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">R2</font></span>)</span> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">R2</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R2</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E4:42_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">R2</font></span>)</span> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i>, , <a class="ref" href="normsp_1.html#T35">NORMSP_1:35</a></i>;<br/><b>thus </b> <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">R2</font></span>)</span> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="normsp_1.html#T43">NORMSP_1:43</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:42"/>
<font color="Maroon">R1</font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">R2</font> is   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/>


</div>
