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<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">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">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">U</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">f1</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>









<b>assume </b><a NAME="E1:17"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f1</font> <a href="ndiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font>
 ;<br/>

<b>let </b><font color="Maroon">f2</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">T</font>,<font color="Maroon">U</font>;<br/>

<b>assume </b><a NAME="E2:17"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f2</font> <a href="ndiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font>
 ;<br/>

<b>consider </b><font color="Maroon">N1</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E4:17"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">N1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>and </b><br/><a NAME="E5:17"/><i><font color="Green">E42</font></i>: 
 ex <font color="Olive">R1</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> st <br/>( <font color="Olive">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> &amp; <font color="Olive">R1</font> <a href="nfcont_1.html#R1">is_continuous_in</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="ndiff_1.html#T52">NDIFF_1:52</a></i>;<br/>
<b>consider </b><font color="Maroon">N2</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E7:17"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>and </b><br/><a NAME="E8:17"/><i><font color="Green">E44</font></i>: 
 ex <font color="Olive">R2</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">T</font>,<font color="Maroon">U</font> st <br/>( <font color="Olive">R2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">U</font> &amp; <font color="Olive">R2</font> <a href="nfcont_1.html#R1">is_continuous_in</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="ndiff_1.html#T52">NDIFF_1:52</a></i>;<br/>
<b>consider </b><font color="Maroon">R1</font> being   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font><b> such that </b><br/><a NAME="E10:17"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R2</font> being   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">T</font>,<font color="Maroon">U</font><b> such that </b><br/><a NAME="E12:17"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">R2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">U</font> &amp; ( for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E13:17"/>
<font color="Maroon">f1</font> <a href="nfcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="ndiff_1.html#T49">NDIFF_1:49</a></i>;<br/>
<b>then consider </b><font color="Maroon">N3</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E15:17"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T17">NFCONT_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon">N</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E17:17"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N3</font> )
 <b>by </b><i><a class="ref" href="ndiff_1.html#T1">NDIFF_1:1</a></i>;<br/>
<a NAME="E18:17"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E4:17_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N3</font>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:17_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:17"/><i><font color="Green">E54</font></i>: 
 <a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#G1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10">Zero_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lopban_1.html#D15">LOPBAN_1:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">LB1</font> =  <a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E21:17"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">LB1</font> <a href="hidden.html#R1">=</a>  <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">LB1</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>;<br/>
<a NAME="E22:17"/><i><font color="Green">E56</font></i>: 
 <a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">T</font>,<font color="Maroon">U</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#G1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10">Zero_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lopban_1.html#D15">LOPBAN_1:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">LB2</font> =  <a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">T</font>,<font color="Maroon">U</font> <b>by </b><i/>;<br/>
<a NAME="E24:17"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">LB2</font> <a href="hidden.html#R1">=</a>  <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">LB2</font>,<font color="Maroon">T</font>,<font color="Maroon">U</font>
 
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>;<br/>
<b>set </b><font color="Maroon">L1</font> =  <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">L2</font> =  <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font>;<br/>
<b>reconsider </b><font color="Maroon">R0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R1</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span></span>)</span> as   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">U</font> <b>by </b><i>, , </i>;<br/>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/><a NAME="E2:17_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:17_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:17_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T4">NFCONT_1:4</a></i>;<br/><a NAME="E5:17_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:17_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N3</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:17_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 <b>by </b><i/>;<br/><a NAME="E8:17_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E9:17_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">R1</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="ndiff_1.html#D5">NDIFF_1:def 5</a></i>;<br/><a NAME="E10:17_2"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/><a NAME="E11:17_2"/>
<font color="Maroon">R2</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="ndiff_1.html#D5">NDIFF_1:def 5</a></i>;<br/><a NAME="E12:17_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/><a NAME="E13:17_2"/><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E14:17_2"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="lopban_1.html#K13">modetrans</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/><a NAME="E15:17_2"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i>, <a class="ref" href="ndiff_2.html#T1" target="_self">Th1</a>, <a class="ref" href="vfunct_1.html#T38">VFUNCT_1:38</a></i>;<br/><a NAME="E16:17_2"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/><a NAME="E17:17_2"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R2</font>
 <b>by </b><i/>;<br/><i><font color="Green">E75</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
<b>by </b><i>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>
;<br/><i><font color="Green">E76</font></i>: <font color="Maroon">R2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon">R2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_2.html#E1:17"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">R2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="vfunct_1.html#D1">VFUNCT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="partfun2.html#T10">PARTFUN2:10</a></i>
;<br/><a NAME="E20:17_2"/><i><font color="Green">E77</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:17_2"/><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">R1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E79</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R1</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R1</font>
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
<b>by </b><i>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>
;<br/><i><font color="Green">E90</font></i>: <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, <a class="txt" href="ndiff_2.html#E5:17"><i><font color="Green">E42</font></i></a>, <a class="ref" href="partfun2.html#T10">PARTFUN2:10</a></i>
;<br/><i><font color="Green">E91</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p5">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R1</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">R2</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p5">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span></span>)</span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
<b>by </b><i><a class="txt" href="ndiff_2.html#E2:17"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ndiff_2.html#E7:17"><i><font color="Green">E43</font></i></a>, <a class="ref" href="vfunct_1.html#D1">VFUNCT_1:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>

;<br/><a NAME="E25:17_2"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="ndiff_2.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="lopban_1.html#K17">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_2.html#E1:17"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="partfun2.html#T6">PARTFUN2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="partfun2.html#T6">PARTFUN2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></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">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p5">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p5">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p5">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R2</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p5">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="ndiff_2.html#E4:17"><i><font color="Green">E40</font></i></a>, <a class="ref" href="lopban_1.html#D5">LOPBAN_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p5">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p0">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">U</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R2</font> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <span class="p0">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">R1</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_2.html#E8:17"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="ndiff_2.html#K2" target="_self">*</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_2.html#E10:17"><i><font color="Green">E45</font></i></a>, <a class="txt" href="ndiff_2.html#E12:17"><i><font color="Green">E46</font></i></a>, <a class="ref" href="vfunct_1.html#D1">VFUNCT_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E27:17"/>
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> <a href="ndiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="ndiff_1.html#D6">NDIFF_1:def 6</a></i>;<br/>

<b>hence </b><a NAME="E28:17"/>
 <a href="ndiff_1.html#K3">diff</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f2</font>,<span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="ndiff_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="ndiff_1.html#D7">NDIFF_1:def 7</a></i>;<br/>


</div>
