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<b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Y</font> is <a href="rcomp_1.html#V1">compact</a>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:11"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<b>assume </b><a NAME="E4:11"/>
not <font color="Maroon">f</font> <a href="fcont_2.html#R1" target="_self">is_uniformly_continuous_on</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:11"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E7:11"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="real_1.html#K5">-</a> <font color="Olive">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; not  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:11"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds 0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  ex <font color="Olive">x2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> ) );<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:11_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:11_1"/><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">x2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <font color="Olive">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>take </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>;<br/><b>thus </b><a NAME="E5:11_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x1</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">s1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E11:11"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="rcomp_1.html#S1">RCOMP_1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> );<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>consider </b><font color="Maroon">x2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:11_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>take </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font>;<br/><b>thus </b><a NAME="E4:11_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x2</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">s2</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E14:11"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="rcomp_1.html#S1">RCOMP_1:sch 1</a>();<br/></i>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:11_3"/><i><font color="Green">E54</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:11_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/><a NAME="E4:11_3"/><i><font color="Green">E60</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a> 
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E5:11_3"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:11_3"/><b>then </b>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:11_3"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E8:11_3"/><b>then </b><i><font color="Green">E61</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T218">XCMPLX_1:218</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E9:11_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E10:11_3"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:11_3"/><b>then </b><i><font color="Green">E63</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T120">XREAL_1:120</a></i>;<br/><a NAME="E12:11_3"/><i><font color="Green">E64</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>;<br/><a NAME="E13:11_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E14:11_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E15:11_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:11"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E17:11"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font>
 ;<br/><a NAME="E2:11_4"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/><b>hence </b><a NAME="E3:11_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E19:11"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">q1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E21:11"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">q1</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">s1</font> &amp; <font color="Maroon">q1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D3">RCOMP_1:def 3</a></i>;<br/>
<a NAME="E22:11"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="fcont_1.html#R1">is_continuous_in</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, , <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">Ns1</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E24:11"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">q1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">s1</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font>
 <b>by </b><i>, <a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E25:11"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E26:11"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/>
<a NAME="E27:11"/><i><font color="Green">E75</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , , <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/>
<b>set </b><font color="Maroon">q2</font> = <font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span>;<br/>
<a NAME="E28:11"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/>
<i><font color="Green">E77</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">q1</font></span>)</span> <a href="real_1.html#K5">-</a> 0
<b>by </b><i>, , , <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">q1</font>

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E31:11"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">s2</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E32:11"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">s2</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s2</font>
 ;<br/><a NAME="E2:11_7"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/><b>hence </b><a NAME="E3:11_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E34:11"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E35:11"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T11">RFUNCT_2:11</a></i>;<br/>
<a NAME="E36:11"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:11"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E38:11"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E39:11"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E40:11"/><b>then </b><i><font color="Green">E83</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">q1</font> is <a href="seq_2.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">q1</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="fcont_1.html#D1">FCONT_1:def 1</a></i>;<br/>
<a NAME="E41:11"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T11">RFUNCT_2:11</a></i>;<br/>
<a NAME="E42:11"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E43:11"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E44:11"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E45:11"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E46:11"/><b>then </b><i><font color="Green">E93</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="fcont_1.html#D1">FCONT_1:def 1</a></i>;<br/>
<a NAME="E47:11"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">q1</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/>
<i><font color="Green">E95</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">q1</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">q1</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
0

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:11_9"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">q1</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="fcont_2.html#D1" target="_self">Def1</a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><a NAME="E3:11_9"/><i><font color="Green">E97</font></i>: 
 <a href="complex1.html#K18">abs</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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">q1</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="fcont_2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:11_9"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">q1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/><a NAME="E5:11_9"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/> <a href="complex1.html#K18">abs</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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">q1</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> = 
 <a href="complex1.html#K18">abs</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">q1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="fcont_2.html#E2:11"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q1</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">s2</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="fcont_2.html#E3:11"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s2</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>
;<br/><b>hence </b><a NAME="E7:11_9"/>
contradiction
 <b>by </b><i>, <a class="txt" href="fcont_2.html#E1:11"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E50:11"/>
contradiction
 ;<br/>


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