<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K2">COMPLEX</a> , <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>thus </b><a NAME="E1:55"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>1</sub></font> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:55_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cfcont_1.html#D2" target="_self">Def2</a></i>;<br/>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E3:55_1"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>(  not 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; not <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> ) ) ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ] means ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&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="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> );<br/>
<a NAME="E4:55_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:55_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></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/>
<a NAME="E2:55_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="E3:55_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E5:55_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:55_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E6:55_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="cfcont_1.html#E5:55_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E6:55_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="cfcont_1.html#S1" target="_self">CFCONT_1:sch 1</a>(<a class="txt" href="cfcont_1.html#E4:55_1"><i><font color="Green">E38</font></i></a>);<br/></i>
<a NAME="E7:55_1"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:55_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="cfcont_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:55_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:55_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:55_1_3"/>
not <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:55_1"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E2:55_1_3"/>
not <span class="p1"><a href="complex1.html#K17">|.</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">c<sub>2</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E7:55_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:55_1_4"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:55_1_4"/><b>then </b><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></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/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:55_1_4"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E5:55_1_4"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E6:55_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="cfcont_1.html#E4:55_1_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:55_1_4"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:55_1_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E8:55_1_4"/><b>then </b><i><font color="Green">E46</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T218">XCMPLX_1:218</a></i>;<br/><a NAME="E9:55_1_4"/><i><font color="Green">E47</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></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/><a NAME="E10:55_1_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="cfcont_1.html#E5:55_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:55_1_4"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E9:55_1_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T120">XREAL_1:120</a></i>;<br/><a NAME="E12:55_1_4"/><b>then </b><i><font color="Green">E48</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E8:55_1_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:55_1_4"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:55_1"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E14:55_1_4"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E12:55_1_4"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:55_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#D4">COMSEQ_2:def 4</a></i>;<br/>
<a NAME="E11:55_1"/><b>then </b>
 <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E9:55_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/>
<a NAME="E12:55_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E1:55_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="cfcont_1.html#E7:55_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cfcont_1.html#E10:55_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cfcont_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:55_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</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">c<sub>2</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E3:55_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/>
<a NAME="E15:55_1"/>
<span class="p1"><a href="complex1.html#K17">|.</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">c<sub>2</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="cfcont_1.html#E14:55_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:55_1"/>
contradiction
 <b>by </b><i><a class="txt" href="cfcont_1.html#E8:55_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:55"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:55_2"/><i><font color="Green">E37</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:55_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:55_2_1"/><i><font color="Green">E39</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:55"><i><font color="Green">E36</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:55_2_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:55_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cfcont_1.html#E3:55_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E6:55_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E7:55_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="cfcont_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E8:55_2_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E5:55_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cfcont_1.html#E6:55_2_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E9:55_2_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:55_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cfcont_1.html#E3:55_2_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cfcont_1.html#E7:55_2_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E10:55_2_1"/>
<span class="p1"><a href="complex1.html#K17">|.</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">c<sub>2</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:55_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cfcont_1.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><a NAME="E3:55_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> is <a href="comseq_2.html#V2">convergent</a>
 <b>by </b><i><a class="ref" href="comseq_2.html#D4">COMSEQ_2:def 4</a></i>;<br/><b>hence </b><a NAME="E4:55_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cfcont_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:55_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:55"/>
<font color="Maroon">c<sub>2</sub></font> <a href="cfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:55"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cfcont_1.html#D2" target="_self">Def2</a></i>;<br/>


</div>
