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<b>let </b><font color="Maroon" title="c1">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E35">A1</font></i>: 
( ( <font color="Maroon" title="c2">seq</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">seq</font> ) or ( <font color="Maroon" title="c2">seq</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp;  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font> ) )
 ;<br/>

<div><i><font color="Green" title="E36">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:50_1"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>and </b><br/><a NAME="E2:50_1"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">seq</font>
 ;<br/><b>consider </b><font color="Maroon" title="c3">g1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E4:50_1"/><i><font color="Green" title="E39">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">g1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:50_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="cfcont_1.html#T34" target="_self" title="CFCONT_1:th.34">Th34</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">g2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E6:50_1"/><i><font color="Green" title="E40">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g2</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:50_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="cfcont_1.html#D4" target="_self" title="CFCONT_1:def.4">Def4</a></i>;<br/><a NAME="E7:50_1"/><i><font color="Green" title="E41">A7</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:50_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="cfcont_1.html#E4:50_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E8:50_1"/><i><font color="Green" title="E42">A8</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:50_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="cfcont_1.html#T48" target="_self" title="CFCONT_1:th.48">Th48</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:50_1_1"/><i><font color="Green" title="E43">A9</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font>
 ;<br/><b>take </b><font color="Maroon" title="c6">n</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>let </b><font color="Maroon" title="c7">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E2:50_1_1"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 ;<br/><a NAME="E3:50_1_1"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E4:50_1_1"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">seq</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T2" title="COMSEQ_1:th.2">COMSEQ_1:2</a></i>;<br/><a NAME="E5:50_1_1"/><b>then </b>
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">seq</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:50_1_1"/><b>then </b>
<font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">seq</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:50_1"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/><a NAME="E7:50_1_1"/><b>then </b>
<font color="Maroon" title="c4">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E4:50_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="cfcont_1.html#E7:50_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E8:50_1_1"/><b>then </b>
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E6:50_1"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E9:50_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c1">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:50_1_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="complex1.html#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:50_1"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E8:50_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="comseq_2.html#D5" title="COMSEQ_2:def.5">COMSEQ_2:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:50_2"/><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>and </b><br/><a NAME="E2:50_2"/><i><font color="Green" title="E38">A11</font></i>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 ;<br/><b>consider </b><font color="Maroon" title="c3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:50_2"/><i><font color="Green" title="E39">A12</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:50_2"><i><font color="Green" title="E38">A11</font></i></a></i>;<br/><a NAME="E5:50_2"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E6:50_2"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">seq</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T2" title="COMSEQ_1:th.2">COMSEQ_1:2</a></i>;<br/><b>hence </b><a NAME="E7:50_2"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E2:50"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="cfcont_1.html#E1:50_2"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="cfcont_1.html#E4:50_2"><i><font color="Green" title="E39">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="cfcont_1.html#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="cfcont_1.html#E2:50"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/>


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