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<b>let </b><font color="Maroon" title="c3">r</font>, <font color="Maroon" title="c4">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">x0</font>, <font color="Olive" title="b2">y0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  ex <font color="Olive" title="b3">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Olive" title="b1">x0</font> st <br/>( <font color="Olive" title="b3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> &amp;  ex <font color="Olive" title="b4">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b5">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b6">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">x0</font></span>)</span></span>)</span> ) ) ) ) &amp;  ex <font color="Olive" title="b1">x0</font>, <font color="Olive" title="b2">y0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  ex <font color="Olive" title="b3">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Olive" title="b1">x0</font> st <br/>( <font color="Olive" title="b3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> &amp;  ex <font color="Olive" title="b4">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b5">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b6">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">x0</font></span>)</span></span>)</span> ) ) ) ) implies <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6_1_2"/><i><font color="Green" title="E8">A6</font></i>: 
 ex <font color="Olive" title="b1">x0</font>, <font color="Olive" title="b2">y0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  ex <font color="Olive" title="b3">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Olive" title="b1">x0</font> st <br/>( <font color="Olive" title="b3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> &amp;  ex <font color="Olive" title="b4">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b5">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b6">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">x0</font></span>)</span></span>)</span> ) ) ) )
 <b>and </b><br/><a NAME="E2:6_1_2"/><i><font color="Green" title="E9">A7</font></i>: 
 ex <font color="Olive" title="b1">x0</font>, <font color="Olive" title="b2">y0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  ex <font color="Olive" title="b3">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Olive" title="b1">x0</font> st <br/>( <font color="Olive" title="b3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> &amp;  ex <font color="Olive" title="b4">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b5">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b6">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">x0</font></span>)</span></span>)</span> ) ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">x0</font>, <font color="Maroon" title="c6">y0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:6_1_2"/><i><font color="Green" title="E10">A8</font></i>: 
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">x0</font>,<font color="Maroon" title="c6">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  ex <font color="Olive" title="b1">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x0</font> st <br/>( <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> &amp;  ex <font color="Olive" title="b2">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b3">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b4">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="#E1:6_1_2"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">N</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x0</font><b> such that </b><br/><a NAME="E6:6_1_2"/><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> &amp;  ex <font color="Olive" title="b1">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b2">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b3">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="#E4:6_1_2"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">L</font> being   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a>, <font color="Maroon" title="c9">R</font> being   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a><b> such that </b><br/><a NAME="E8:6_1_2"/><i><font color="Green" title="E12">A10</font></i>: 
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E6:6_1_2"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">y1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:6_1_2"/><i><font color="Green" title="E13">A11</font></i>: 
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  ex <font color="Olive" title="b1">N</font> being   <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c10">x1</font> st <br/>( <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> &amp;  ex <font color="Olive" title="b2">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b3">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b4">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">x1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">x1</font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="#E2:6_1_2"><i><font color="Green" title="E9">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">N1</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c10">x1</font><b> such that </b><br/><a NAME="E12:6_1_2"/><i><font color="Green" title="E14">A12</font></i>: 
( <font color="Maroon" title="c12">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p2">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> &amp;  ex <font color="Olive" title="b1">L</font> being  <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a> ex <font color="Olive" title="b2">R</font> being  <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a> st <br/>( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b3">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N1</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">x1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">x1</font></span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="#E10:6_1_2"><i><font color="Green" title="E13">A11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">L1</font> being   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LINEAR</a>, <font color="Maroon" title="c14">R1</font> being   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">REST</a><b> such that </b><br/><a NAME="E14:6_1_2"/><i><font color="Green" title="E15">A13</font></i>: 
( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">L1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N1</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">L1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">x1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">x1</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E12:6_1_2"><i><font color="Green" title="E14">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E16:6_1_2"/><i><font color="Green" title="E16">A14</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c8">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="ref" href="fdiff_1.html#D4" title="FDIFF_1:def.4">FDIFF_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">p1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E18:6_1_2"/><i><font color="Green" title="E17">A15</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c13">L1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="ref" href="fdiff_1.html#D4" title="FDIFF_1:def.4">FDIFF_1:def 4</a></i>;<br/>
<a NAME="E19:6_1_2"/><i><font color="Green" title="E18">A16</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 
<b>by </b><i><a class="txt" href="#E8:6_1_2"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="#E16:6_1_2"><i><font color="Green" title="E16">A14</font></i></a></i>;<br/>
<a NAME="E20:6_1_2"/><i><font color="Green" title="E19">A17</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 
<b>by </b><i><a class="txt" href="#E14:6_1_2"><i><font color="Green" title="E15">A13</font></i></a>, <a class="txt" href="#E18:6_1_2"><i><font color="Green" title="E17">A15</font></i></a></i>;<br/>
<a NAME="E21:6_1_2"/><i><font color="Green" title="E20">A18</font></i>: 
( <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> &amp; <font color="Maroon" title="c6">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y1</font> )
 
<b>by </b><i><a class="txt" href="#E4:6_1_2"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="#E10:6_1_2"><i><font color="Green" title="E13">A11</font></i></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></i>;<br/>
<div><i><font color="Green" title="E21">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> )</span><br/><b>assume </b><a NAME="E1:6_1_2_1"/><i><font color="Green" title="E22">A20</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N1</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"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span></span><br/><a NAME="E2:6_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_2.html#K1" title="PDIFF_2:func.1">SVF1</a> <span class="p3">(<span class="default"><a href="pdiff_3.html#K2" title="PDIFF_3:func.2">pdiff2</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">z</font></span>)</span>,<font color="Maroon" title="c2">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:6_1_2"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/><a NAME="E3:6_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">L1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:6_1_2"><i><font color="Green" title="E15">A13</font></i></a>, <a class="txt" href="#E21:6_1_2"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="#E1:6_1_2_1"><i><font color="Green" title="E22">A20</font></i></a></i>;<br/><a NAME="E4:6_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">L1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:6_1_2"><i><font color="Green" title="E16">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E5:6_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E18:6_1_2"><i><font color="Green" title="E17">A15</font></i></a>, <a class="txt" href="#E19:6_1_2"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="#E20:6_1_2"><i><font color="Green" title="E19">A17</font></i></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>consider </b><font color="Maroon" title="c17">N0</font> being    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c5">x0</font><b> such that </b><br/><a NAME="E24:6_1_2"/><i><font color="Green" title="E22">A21</font></i>: 
( <font color="Maroon" title="c17">N0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c17">N0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">N1</font> )
 <b>by </b><i><a class="txt" href="#E21:6_1_2"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="rcomp_1.html#T38" title="RCOMP_1:th.38">RCOMP_1:38</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E26:6_1_2"/><i><font color="Green" title="E23">A22</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">g</font> &amp; <font color="Maroon" title="c17">N0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7" title="RCOMP_1:def.7">RCOMP_1:def 7</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c18">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c19">s1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E28:6_1_2"/><i><font color="Green" title="E24">A23</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c19">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E1:6_1_2_2"/>
<font color="Maroon" title="c18">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E26:6_1_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/><b>hence </b><a NAME="E2:6_1_2_2"/>
<font color="Maroon" title="c19">s1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E28:6_1_2"><i><font color="Green" title="E24">A23</font></i></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>
<a NAME="E30:6_1_2"/><b>then </b><i><font color="Green" title="E25">A24</font></i>: 
<font color="Maroon" title="c19">s1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-zero</a>
 
<b>by </b><i><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/>
<a NAME="E31:6_1_2"/>
( <font color="Maroon" title="c19">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c19">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E28:6_1_2"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="seq_4.html#T46" title="SEQ_4:th.46">SEQ_4:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">h</font> = <font color="Maroon" title="c19">s1</font> as  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="txt" href="#E30:6_1_2"><i><font color="Green" title="E25">A24</font></i></a>, <a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E33:6_1_2"/><i><font color="Green" title="E26">A25</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N1</font> &amp; <font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N1</font> &amp; <font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font> )</span><br/>

<a NAME="E1:6_1_2_3"/><i><font color="Green" title="E27">A26</font></i>: 
<font color="Maroon" title="c18">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E26:6_1_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/>
<a NAME="E2:6_1_2_3"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:6_1_2_3"/><b>then </b>
<font color="Maroon" title="c18">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
 
<b>by </b><i><a class="txt" href="#E26:6_1_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="xreal_1.html#T78" title="XREAL_1:th.78">XREAL_1:78</a></i>;<br/>
<a NAME="E4:6_1_2_3"/><b>then </b><i><font color="Green" title="E28">A27</font></i>: 
<font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">g</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:6_1_2_3"/><i><font color="Green" title="E29">AA</font></i>: 
<font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c18">g</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E26:6_1_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="txt" href="#E1:6_1_2_3"><i><font color="Green" title="E27">A26</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:6_1_2_3"/><i><font color="Green" title="E30">A28</font></i>: 
<font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="#E4:6_1_2_3"><i><font color="Green" title="E28">A27</font></i></a>, <a class="txt" href="#E5:6_1_2_3"><i><font color="Green" title="E29">AA</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c5">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N1</font> &amp; <font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font> )</span><br/>

<b>thus </b><a NAME="E7:6_1_2_3"/>
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N1</font> &amp; <font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font> )
 <b>by </b><i><a class="txt" href="#E24:6_1_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="#E26:6_1_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="txt" href="#E28:6_1_2"><i><font color="Green" title="E24">A23</font></i></a>, <a class="txt" href="#E6:6_1_2_3"><i><font color="Green" title="E30">A28</font></i></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>
<div><i><font color="Green" title="E27">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span><br/><a NAME="E1:6_1_2_4"/><i><font color="Green" title="E28">X</font></i>: 
<font color="Maroon" title="c21">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:6_1_2_4"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">N1</font> &amp; <font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">x0</font> )
 <b>by </b><i><a class="txt" href="#E33:6_1_2"><i><font color="Green" title="E26">A25</font></i></a></i>;<br/><a NAME="E3:6_1_2_4"/><b>then </b><i><font color="Green" title="E29">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E22:6_1_2"><i><font color="Green" title="E21">A19</font></i></a></i>;<br/><a NAME="E4:6_1_2_4"/>
<font color="Maroon" title="c20">h</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-zero</a>
 <b>by </b><i><a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></i>;<br/><a NAME="E5:6_1_2_4"/><b>then </b><i><font color="Green" title="E30">A31</font></i>: 
<font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:6_1_2_4"><i><font color="Green" title="E28">X</font></i></a>, <a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/><a NAME="E6:6_1_2_4"/><i><font color="Green" title="E31">A32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:6_1_2_4"><i><font color="Green" title="E29">A30</font></i></a>, <a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>;<br/><i><font color="Green" title="E32">A33</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> = 
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
<b>by </b><i><a class="txt" href="#E5:6_1_2_4"><i><font color="Green" title="E30">A31</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c3">r</font>

;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> = 
<font color="Maroon" title="c4">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon" title="c4">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
<b>by </b><i><a class="txt" href="#E5:6_1_2_4"><i><font color="Green" title="E30">A31</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c4">s</font>

;<br/><a NAME="E9:6_1_2_4"/><b>then </b><i><font color="Green" title="E33">A34</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:6_1_2_4"><i><font color="Green" title="E31">A32</font></i></a>, <a class="txt" href="#E7:6_1_2_4"><i><font color="Green" title="E32">A33</font></i></a>, <a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>;<br/><a NAME="E10:6_1_2_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/><a NAME="E11:6_1_2_4"/><b>then </b><i><font color="Green" title="E34">A35</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">R</font>
 ;<br/><a NAME="E12:6_1_2_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/><a NAME="E13:6_1_2_4"/><b>then </b><i><font color="Green" title="E35">A36</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">R1</font>
 ;<br/><i><font color="Green" title="E36">A37</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T10" title="VALUED_1:th.10">VALUED_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:6_1_2_4"><i><font color="Green" title="E34">A35</font></i></a>, <a class="ref" href="funct_2.html#T185" title="FUNCT_2:th.185">FUNCT_2:185</a>, <a class="txt" href="#E1:6_1_2_4"><i><font color="Green" title="E28">X</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T10" title="VALUED_1:th.10">VALUED_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E13:6_1_2_4"><i><font color="Green" title="E35">A36</font></i></a>, <a class="ref" href="funct_2.html#T185" title="FUNCT_2:th.185">FUNCT_2:185</a>, <a class="txt" href="#E1:6_1_2_4"><i><font color="Green" title="E28">X</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
;<br/><a NAME="E16:6_1_2_4"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</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"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:6_1_2_4"><i><font color="Green" title="E33">A34</font></i></a>, <a class="txt" href="#E14:6_1_2_4"><i><font color="Green" title="E36">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E17:6_1_2_4"/>
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">n</font>
 <b>by </b><i><a class="ref" href="valued_1.html#T15" title="VALUED_1:th.15">VALUED_1:15</a>, <a class="txt" href="#E1:6_1_2_4"><i><font color="Green" title="E28">X</font></i></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>
<a NAME="E35:6_1_2"/><b>then </b><i><font color="Green" title="E28">A38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
<b>by </b><i><a class="ref" href="valued_0.html#D18" title="VALUED_0:def.18">VALUED_0:def 18</a></i>;<br/>
<a NAME="E36:6_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="txt" href="#E34:6_1_2"><i><font color="Green" title="E27">A29</font></i></a></i>;<br/>
<a NAME="E37:6_1_2"/><b>then </b><i><font color="Green" title="E29">A39</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="txt" href="#E35:6_1_2"><i><font color="Green" title="E28">A38</font></i></a>, <a class="ref" href="seq_4.html#T40" title="SEQ_4:th.40">SEQ_4:40</a></i>;<br/>
<a NAME="E38:6_1_2"/><i><font color="Green" title="E30">A40</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D3" title="FDIFF_1:def.3">FDIFF_1:def 3</a></i>;<br/>
<a NAME="E39:6_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D3" title="FDIFF_1:def.3">FDIFF_1:def 3</a></i>;<br/>
<a NAME="E40:6_1_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E37:6_1_2"><i><font color="Green" title="E29">A39</font></i></a>, <a class="txt" href="#E38:6_1_2"><i><font color="Green" title="E30">A40</font></i></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></i>;<br/>
<b>hence </b><a NAME="E41:6_1_2"/>
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
