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<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x0</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Maroon" title="c3">f</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x0</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x0</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font> implies <font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">x0</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:64"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="cfdiff_1.html#R1" title="CFDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">x0</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">N</font> being    <a href="cfdiff_1.html#M1" title="CFDIFF_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x0</font><span class="kw"> such that </span><br/><a NAME="E3:64"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E4:64"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="cfdiff_1.html#NM2" title="CFDIFF_1:NM.2">C_LINEAR</a> ex <font color="Olive" title="b2">R</font> being  <a href="cfdiff_1.html#NM1" title="CFDIFF_1:NM.1">C_REST</a> st <br/>for <font color="Olive" title="b3">x</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D6" target="_self" title="CFDIFF_1:def.6">Def6</a></span>;<br/>
<a NAME="E5:64"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#T7" target="_self" title="CFDIFF_1:th.7">Th7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:64_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <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="c6">g</font>
 <span class="kw">and </span><br/><a NAME="E3:64_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> : <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">x0</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="c6">g</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D5" target="_self" title="CFDIFF_1:def.5">Def5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">s2</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c4">x0</font> as   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> ;<br/><span class="kw">let </span><font color="Maroon" title="c8">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c8">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> &amp; ( 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="c8">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> ) implies ( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">s1</font></span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:64_1"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E6:64_1"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c8">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E7:64_1"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 <span class="kw">and </span><br/><a NAME="E8:64_1"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
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="c8">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">s1</font></span>)</span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c9">l</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> <span class="kw"> such that </span><br/><a NAME="E10:64_1"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
for <font color="Olive" title="b1">m</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="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><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="c8">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">x0</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="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E7:64_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="comseq_2.html#D5" title="COMSEQ_2:def.5">COMSEQ_2:def 5</a></span>;<br/><span class="lab"><font color="Green" title="E43">A11</font></span>:  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c7">s2</font> = 
<font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T50" title="CFCONT_1:th.50">CFCONT_1:50</a></span>
<br/>.= 
<font color="Maroon" title="c4">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <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> ) <span class="kw">-&gt; </span>   <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>  = <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">s3</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a><span class="kw"> such that </span><br/><a NAME="E13:64_1"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
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="c10">s3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span><a NAME="E14:64_1"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
<font color="Maroon" title="c10">s3</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c7">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:64_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="cfcont_1.html#T2" title="CFCONT_1:th.2">CFCONT_1:2</a></span>;<br/><a NAME="E15:64_1"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c7">s2</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T48" title="CFCONT_1:th.48">CFCONT_1:48</a></span>;<br/><a NAME="E16:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A15</font></span>: 
<font color="Maroon" title="c10">s3</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E14:64_1"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="comseq_2.html#T25" title="COMSEQ_2:th.25">COMSEQ_2:25</a></span>;<br/> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c10">s3</font> = 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c7">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:64_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="cfcont_1.html#T2" title="CFCONT_1:th.2">CFCONT_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c4">x0</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E7:64_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E15:64_1"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E11:64_1"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="ref" href="comseq_2.html#T26" title="COMSEQ_2:th.26">COMSEQ_2:26</a></span>
<br/>.= 
 <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 

;<br/><a NAME="E18:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A16</font></span>: 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s3</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:64_1"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="cfcont_1.html#T43" title="CFCONT_1:th.43">CFCONT_1:43</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c7">s2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font> as  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V8</a>() <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> ;<br/><div><span class="lab"><font color="Green" title="E49">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c12">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> <span class="kw"> such that </span><a NAME="E2:64_1_3"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
<font color="Maroon" title="c10">s3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:64_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:64_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E2:64_1_3"><span class="lab"><font color="Green" title="E50">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E50">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c12">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> <span class="kw"> such that </span><a NAME="E2:64_1_4"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s3</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:64_1_4"/>
<font color="Maroon" title="c10">s3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_4"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:64_1"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E22:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A21</font></span>: 
<font color="Maroon" title="c10">s3</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_1.html#T4" title="COMSEQ_1:th.4">COMSEQ_1:4</a></span>;<br/><a NAME="E23:64_1"/>
<font color="Maroon" title="c10">s3</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:64_1"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="cfcont_1.html#T43" title="CFCONT_1:th.43">CFCONT_1:43</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">h</font> = <font color="Maroon" title="c10">s3</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font> as   <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:64_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E22:64_1"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="ref" href="cfdiff_1.html#D1" target="_self" title="CFDIFF_1:def.1">Def1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">n</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T2" title="CFCONT_1:th.2">CFCONT_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E26:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">n</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="cfdiff_1.html#T18" target="_self" title="CFDIFF_1:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T2" title="CFCONT_1:th.2">CFCONT_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A23</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:64_1"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="valued_0.html#T27" title="VALUED_0:th.27">VALUED_0:27</a></span>
;<br/><a NAME="E29:64_1"/><span class="lab"><font color="Green" title="E54">A24</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">c</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="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:64_1_8"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">c</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">c</font> or <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_8_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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> <span class="kw"> such that </span><br/><a NAME="E3:64_1_8_1"/><span class="lab"><font color="Green" title="E55">A25</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T190" title="FUNCT_2:th.190">FUNCT_2:190</a></span>;<br/>
<a NAME="E4:64_1_8_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_8_1"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>;<br/>
<a NAME="E5:64_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:64_1_8_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:64_1_8"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">c</font></span><br/>

<a NAME="E3:64_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A26</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<font color="Maroon" title="c11">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c13">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_8"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:64_1_8"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E55">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="c13">p</font> implies  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 <br/>for <font color="Olive" title="b2">m</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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></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="c13">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_9"/><span class="lab"><font color="Green" title="E56">A28</font></span>: 
 <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="c13">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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 <br/>for <font color="Olive" title="b2">m</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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></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="c13">p</font></span><br/><span class="kw">take </span><font color="Maroon" title="c14">n</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</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="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></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="c13">p</font></span><br/><span class="kw">thus </span><a NAME="E2:64_1_9"/>
for <font color="Olive" title="b1">m</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="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></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="c13">p</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> implies <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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></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="c13">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_9_1"/>
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></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="c13">p</font></span><br/>

<a NAME="E2:64_1_9_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<span class="kw">then </span><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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
<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="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64_1"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="ref" href="funct_2.html#T186" title="FUNCT_2:th.186">FUNCT_2:186</a></span>
<br/>.= 
<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="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
<br/>.= 
<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="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:64_1_9_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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></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="c13">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_9"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E31:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A29</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_2.html#D4" title="COMSEQ_2:def.4">COMSEQ_2:def 4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">n</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span><br/><a NAME="E1:64_1_10"/><span class="lab"><font color="Green" title="E57">A30</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:64_1"><span class="lab"><font color="Green" title="E50">A19</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" title="VALUED_1:th.10">VALUED_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span>

<br/>.= 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_10"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E33:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A31</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/><a NAME="E34:64_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span> or <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_11"/><span class="lab"><font color="Green" title="E58">A32</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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> <span class="kw"> such that </span><br/><a NAME="E3:64_1_11"/><span class="lab"><font color="Green" title="E59">A33</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T190" title="FUNCT_2:th.190">FUNCT_2:190</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c13">y</font> as   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_11"><span class="lab"><font color="Green" title="E58">A32</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c9">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="cfdiff_1.html#T18" target="_self" title="CFDIFF_1:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfcont_1.html#T2" title="CFCONT_1:th.2">CFCONT_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c8">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">n</font></span>)</span>

;<br/>
<a NAME="E6:64_1_11"/><span class="kw">then </span>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">x0</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="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E7:64_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c15">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> : <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">x0</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="c6">g</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_11"><span class="lab"><font color="Green" title="E59">A33</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:64_1_11"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E35:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E29:64_1"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="ref" href="cfdiff_1.html#T22" target="_self" title="CFDIFF_1:th.22">Th22</a></span>;<br/><a NAME="E36:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A35</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:64_1"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="ref" href="comseq_2.html#T29" title="COMSEQ_2:th.29">COMSEQ_2:29</a></span>;<br/><a NAME="E37:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A36</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:64_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="comseq_2.html#T13" title="COMSEQ_2:th.13">COMSEQ_2:13</a></span>;<br/><span class="kw">hence </span><a NAME="E38:64_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="ref" href="cfcont_1.html#T44" title="CFCONT_1:th.44">CFCONT_1:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">s1</font></span>)</span></span><br/> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:64_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E35:64_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="comseq_2.html#T30" title="COMSEQ_2:th.30">COMSEQ_2:30</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">then </span> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span> = 
<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:64_1"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="txt" href="#E36:64_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E31:64_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="comseq_2.html#T14" title="COMSEQ_2:th.14">COMSEQ_2:14</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:64_1"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="txt" href="#E31:64_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="comseq_2.html#D5" title="COMSEQ_2:def.5">COMSEQ_2:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E41:64_1"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:64_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E28:64_1"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="ref" href="cfcont_1.html#T45" title="CFCONT_1:th.45">CFCONT_1:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:64"/>
<font color="Maroon" title="c3">f</font> <a href="cfcont_1.html#R1" title="CFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="cfcont_1.html#T53" title="CFCONT_1:th.53">CFCONT_1:53</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
