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<b>let </b><font color="Maroon">CNS1</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">CNS2</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">CNS1</font>,<font color="Maroon">CNS2</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">CNS1</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">Y</font> is <a href="ncfcont1.html#V1">compact</a>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R7">is_continuous_on</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E3:34"/><i><font color="Green">E52</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="ncfcont1.html#D21">NCFCONT1:def 21</a></i>;<br/>
<b>assume </b><a NAME="E4:34"/>
not <font color="Maroon">f</font> <a href="ncfcont2.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="E6:34"/><i><font color="Green">E53</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E7:34"/><i><font color="Green">E54</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="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font> 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; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">x2</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; not <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:34_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T21">NAT_1:21</a></i>;<br/><a NAME="E2:34_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><b>hence </b><a NAME="E3:34_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><a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font>] <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="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font> st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">x2</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></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 <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> ) );<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:34_2"/>
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="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font><b> such that </b><br/><a NAME="E3:34_2"/><i><font color="Green">E57</font></i>: 
 ex <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font> 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; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">x2</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></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 <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>;<br/><b>thus </b><a NAME="E4:34_2"/>
<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="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS1</font><b> such that </b><br/><a NAME="E11:34"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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="normsp_1.html#K2">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font>] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></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 <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> );<br/>
<a NAME="E12:34"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">x2</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s2</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS1</font><b> such that </b><br/><a NAME="E14:34"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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="normsp_1.html#K2">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:34_3"/><i><font color="Green">E64</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:34_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">p</font> <a href="real_1.html#K2">"</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="E4:34_3"/><i><font color="Green">E84</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K2">"</a> 
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E5:34_3"/><i><font color="Green">E85</font></i>: 
<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="E6:34_3"/><b>then </b>
<font color="Maroon">p</font> <a href="real_1.html#K2">"</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="E7:34_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="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E8:34_3"/><b>then </b><i><font color="Green">E86</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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E9:34_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E10:34_3"/><i><font color="Green">E89</font></i>: 
0 <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>, , </i>;<br/><a NAME="E11:34_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="E12:34_3"/><b>then </b><i><font color="Green">E90</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="ncfcont2.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T120">XREAL_1:120</a></i>;<br/><i><font color="Green">E91</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS1</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS1</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/><a NAME="E14:34_3"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></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="E15:34_3"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></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="E16:34_3"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS1</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></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="E16:34"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D15">CLVECT_1:def 15</a></i>;<br/>
<a NAME="E17:34"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS1</font>
 
<b>by </b><i>, <a class="ref" href="clvect_1.html#D17">CLVECT_1:def 17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS1</font>;<br/><b>assume </b><a NAME="E1:34_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="E2:34_4"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">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="normsp_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T8">NCFCONT1:8</a></i>;<br/><b>hence </b><a NAME="E3:34_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="E19:34"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
;<br/>
<a NAME="E20:34"/><b>then </b><i><font color="Green">E95</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="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS1</font><b> such that </b><br/><a NAME="E22:34"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">q1</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">s1</font> &amp; <font color="Maroon">q1</font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">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="ncfcont1.html#D6">NCFCONT1:def 6</a></i>;<br/>
<a NAME="E23:34"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="ncfcont1.html#R1">is_continuous_in</a>  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, , <a class="ref" href="ncfcont1.html#D21">NCFCONT1:def 21</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="E25:34"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">q1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">s1</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font>
 <b>by </b><i>, <a class="ref" href="bhsp_3.html#D4">BHSP_3:def 4</a></i>;<br/>
<a NAME="E26:34"/><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font>
 
<b>by </b><i><a class="ref" href="bhsp_3.html#D4">BHSP_3:def 4</a></i>;<br/>
<a NAME="E27:34"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i>, <a class="ref" href="clopban3.html#T10">CLOPBAN3:10</a></i>;<br/>
<a NAME="E28:34"/><i><font color="Green">E103</font></i>: 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS1</font>
 
<b>by </b><i>, , <a class="ref" href="ncfcont2.html#D1" target="_self">Def1</a>, <a class="ref" href="clopban3.html#T11">CLOPBAN3:11</a></i>;<br/>
<b>set </b><font color="Maroon">q2</font> = <font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span>;<br/>
<a NAME="E29:34"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i>, , <a class="ref" href="clvect_1.html#T116">CLVECT_1:116</a></i>;<br/>
<i><font color="Green">E106</font></i>:  <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">q1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS1</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="clvect_1.html#T122">CLVECT_1:122</a></i>
<br/>.= 
 <a href="clvect_1.html#K11">lim</a> <font color="Maroon">q1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">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="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="normsp_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="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</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="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_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="rlvect_1.html#K6">-</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="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T7">LOPBAN_3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_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="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_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="lopban_3.html#T7">LOPBAN_3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_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="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_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="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_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="clvect_1.html#D12">CLVECT_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_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="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_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> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_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="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_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="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ns1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T7">LOPBAN_3:7</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E32:34"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</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="E33:34"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">s2</font>
 
<b>by </b><i><a class="ref" href="bhsp_3.html#D4">BHSP_3:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_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="E1:34_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="E2:34_7"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">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="normsp_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T8">NCFCONT1:8</a></i>;<br/><b>hence </b><a NAME="E3:34_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="E35:34"/><b>then </b><i><font color="Green">E109</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="E36:34"/>
 <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="ncfcont1.html#T9">NCFCONT1:9</a></i>;<br/>
<a NAME="E37:34"/><b>then </b><i><font color="Green">E110</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="E38:34"/><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="E39:34"/><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="ncfcont2.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E40:34"/><b>then </b><i><font color="Green">E111</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="E41:34"/><b>then </b><i><font color="Green">E112</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="ncfcont1.html#K5">*</a> <font color="Maroon">q1</font> is <a href="clvect_1.html#V6">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="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">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="ncfcont1.html#K5">*</a> <font color="Maroon">q1</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="ncfcont1.html#D15">NCFCONT1:def 15</a></i>;<br/>
<a NAME="E42:34"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</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="ncfcont2.html#D5" target="_self">Def5</a>, <a class="ref" href="ncfcont1.html#T9">NCFCONT1:9</a></i>;<br/>
<a NAME="E43:34"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</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="ncfcont2.html#D6" target="_self">Def6</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E44:34"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</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="E45:34"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</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="ncfcont2.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E46:34"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</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="E47:34"/><b>then </b><i><font color="Green">E115</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="ncfcont1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> is <a href="clvect_1.html#V6">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="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">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="ncfcont1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="ncfcont2.html#D2" target="_self">Def2</a>, <a class="ref" href="ncfcont2.html#D3" target="_self">Def3</a>, <a class="ref" href="ncfcont1.html#D15">NCFCONT1:def 15</a></i>;<br/>
<a NAME="E48:34"/><b>then </b><i><font color="Green">E116</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="ncfcont1.html#K5">*</a> <font color="Maroon">q1</font></span>)</span> <a href="clvect_1.html#K7">-</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="ncfcont1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="txt" href="ncfcont2.html#E2:34"><i><font color="Green">E47</font></i></a>, <a class="ref" href="clvect_1.html#T116">CLVECT_1:116</a></i>;<br/>
<i><font color="Green">E117</font></i>:  <a href="clvect_1.html#K11">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="ncfcont1.html#K5">*</a> <font color="Maroon">q1</font></span>)</span> <a href="clvect_1.html#K7">-</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="ncfcont1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</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="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">q1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont2.html#E2:34"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="clvect_1.html#T122">CLVECT_1:122</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:34_9"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">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/><span class="p1"><a href="clvect_1.html#K5">||.</a><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"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="ncfcont1.html#K5">*</a> <font color="Maroon">q1</font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="ncfcont1.html#K5">*</a> <span class="p0">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="txt" href="ncfcont2.html#E3:34"><i><font color="Green">E52</font></i></a>, <a class="ref" href="clvect_1.html#D17">CLVECT_1:def 17</a></i>;<br/><a NAME="E3:34_9"/><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><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"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="ncfcont1.html#K5">*</a> <font color="Maroon">q1</font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="ncfcont1.html#K5">*</a> <span class="p0">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="ncfcont2.html#E6:34"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E4:34_9"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">q1</font> <a href="normsp_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="ncfcont1.html#T8">NCFCONT1:8</a></i>;<br/><a NAME="E5:34_9"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="normsp_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="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T8">NCFCONT1:8</a></i>;<br/><span class="p1"><a href="clvect_1.html#K5">||.</a><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"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="ncfcont1.html#K5">*</a> <font color="Maroon">q1</font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="ncfcont1.html#K5">*</a> <span class="p0">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><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="ncfcont1.html#K5">*</a> <font color="Maroon">q1</font></span>)</span> <a href="clvect_1.html#K7">-</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="ncfcont1.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><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="ncfcont1.html#K5">*</a> <font color="Maroon">q1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <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="ncfcont1.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><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="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">q1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <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="ncfcont1.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont2.html#E1:34"><i><font color="Green">E46</font></i></a>, <a class="ref" href="ncfcont1.html#T10">NCFCONT1:10</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><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="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">q1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <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="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="ncfcont2.html#T3" target="_self">Th3</a>, <a class="ref" href="ncfcont1.html#T10">NCFCONT1:10</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">q1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <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="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="txt" href="ncfcont2.html#E1:34"><i><font color="Green">E46</font></i></a>, <a class="txt" href="ncfcont2.html#E8:34"><i><font color="Green">E55</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">q1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q1</font> <a href="clvect_1.html#K7">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="ncfcont2.html#T3" target="_self">Th3</a>, <a class="txt" href="ncfcont2.html#E9:34"><i><font color="Green">E56</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<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="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">Ns1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i>, <a class="ref" href="ncfcont2.html#D4" target="_self">Def4</a>, <a class="ref" href="lopban_3.html#T7">LOPBAN_3:7</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<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="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<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></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_3.html#T7">LOPBAN_3:7</a></i>
;<br/><b>hence </b><a NAME="E7:34_9"/>
contradiction
 <b>by </b><i>, <a class="txt" href="ncfcont2.html#E7:34"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E51:34"/>
contradiction
 ;<br/>


</div>
