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<a NAME="E1:8_1_1"/>
 ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">dv</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">h</font> being  <a href="real_1.html#NM1">Real</a>  st  <a href="complex1.html#K18">abs</a> <font color="Olive">h</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">d</font> &amp; <font color="Olive">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">h</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Olive">h</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Olive">dv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> ) ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:8_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">h</font> being  <a href="real_1.html#NM1">Real</a>  st  <a href="complex1.html#K18">abs</a> <font color="Olive">h</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">d</font> &amp; <font color="Olive">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">h</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Olive">h</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> ) ) ) )
 ;<br/>


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