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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>3</sub></font>, <a href="numbers.html#K2">COMPLEX</a> ;<br/>









<b>assume </b><a NAME="E1:78"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:78"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T81" target="_self">Th81</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:78"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><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="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="cfunct_1.html#E1:78"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cfunct_1.html#T81" target="_self">Th81</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:78_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:78_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="cfunct_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:78_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:78_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:78_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="cfunct_1.html#E3:78"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E7:78_1"/><i><font color="Green">E60</font></i>: 
<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="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="cfunct_1.html#E5:78"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cfunct_1.html#E5:78_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E8:78_1"/><i><font color="Green">E61</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E9:78_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <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="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E10:78_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <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="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="cfunct_1.html#E6:78_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="cfunct_1.html#E7:78_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="cfunct_1.html#E8:78_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/><a NAME="E11:78_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>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><b>hence </b><a NAME="E12:78_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>4</sub></font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="cfunct_1.html#E2:78_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cfunct_1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:78"/>
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K3">(#)</a> <font color="Maroon">c<sub>5</sub></font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T81" target="_self">Th81</a></i>;<br/>

<a NAME="E8:78"/>
 <a href="comseq_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cfunct_1.html#E1:78"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cfunct_1.html#T86" target="_self">Th86</a></i>;<br/>
<a NAME="E9:78"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cfunct_1.html#E1:78"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cfunct_1.html#T87" target="_self">Th87</a></i>;<br/>
<b>hence </b><a NAME="E10:78"/>
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>;<br/>


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