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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>; <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">R1</font>, <font color="Olive" title="b2">R2</font> being  <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> holds <br/> ( <font color="Olive" title="b1">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Olive" title="b2">R2</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> &amp; <font color="Olive" title="b1">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">R2</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> )</span><br/><b>let </b><font color="Maroon" title="c3">R1</font>, <font color="Maroon" title="c4">R2</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">R2</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> &amp; <font color="Maroon" title="c3">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</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> )</span><br/>







<a NAME="E1:42"/><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c3">R1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c4">R2</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="E2:42"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c3">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">R2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#T38" title="VFUNCT_1:th.38">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" 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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">R2</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> &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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">R2</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> <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> )</span><br/><a NAME="E1:42_1"/><i><font color="Green" title="E34">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="c3">R1</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="c3">R1</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/><a NAME="E2:42_1"/><i><font color="Green" title="E35">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"><font color="Maroon" title="c4">R2</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">R2</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">A5</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">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">R2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</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#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="ndiff_1.html#T30" target="_self" title="NDIFF_1:th.30">Th30</a></i>
<br/>.= 
<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="c3">R1</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="normsp_1.html#K3" title="NORMSP_1:func.3">+</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">R2</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#T9" target="_self" title="NDIFF_1:th.9">Th9</a></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" 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">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">R2</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#E1:42_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="ndiff_1.html#E2:42_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="normsp_1.html#T34" title="NORMSP_1:th.34">NORMSP_1:34</a></i>; <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> <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">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">R2</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> <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></span><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">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">R2</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> = 
<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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#E1:42_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="ndiff_1.html#E2:42_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="ndiff_1.html#E3:42_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="normsp_1.html#T42" title="NORMSP_1:th.42">NORMSP_1:42</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#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:42"/>
<font color="Maroon" title="c3">R1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">R2</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#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="ndiff_1.html#D5" target="_self" title="NDIFF_1:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</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></span><br/>

<a NAME="E5:42"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c3">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="vfunct_1.html#T38" title="VFUNCT_1:th.38">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" 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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</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> &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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</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> <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> )</span><br/><a NAME="E1:42_2"/><i><font color="Green" title="E35">A7</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="c3">R1</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="c3">R1</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/><a NAME="E2:42_2"/><i><font color="Green" title="E36">A8</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">R2</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">R2</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="E37">A9</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">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">R2</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#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="ndiff_1.html#T30" target="_self" title="NDIFF_1:th.30">Th30</a></i>
<br/>.= 
<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="c3">R1</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="normsp_1.html#K4" title="NORMSP_1:func.4">-</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">R2</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#T12" target="_self" title="NDIFF_1:th.12">Th12</a></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" 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">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</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#E1:42_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="ndiff_1.html#E2:42_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="normsp_1.html#T35" title="NORMSP_1:th.35">NORMSP_1:35</a></i>; <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> <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">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</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> <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></span><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">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</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> = 
<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> <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">T</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:42_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="ndiff_1.html#E2:42_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="ndiff_1.html#E3:42_2"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="normsp_1.html#T43" title="NORMSP_1:th.43">NORMSP_1:43</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#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/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:42"/>
<font color="Maroon" title="c3">R1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">R2</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#E5:42"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="ndiff_1.html#D5" target="_self" title="NDIFF_1:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
