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

<span class="kw">let </span><font color="Maroon" title="c1">q3</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"> for <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  holds <br/><font color="Maroon" title="c1">q3</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Olive" title="b1">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c2">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  implies <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c2">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:109"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c2">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</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="c1">q3</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c2">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">s0</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="c1">q3</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rcomp_1.html#D4" title="RCOMP_1:def.4">RCOMP_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c2">X</font> or  not <font color="Maroon" title="c4">s1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  or  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:109"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E4:109"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c4">s1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c2">X</font></span><br/>

 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">s0</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s0</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="seq_4.html#T59" title="SEQ_4:th.59">SEQ_4:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">q3</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/>
<a NAME="E6:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c1">q3</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">s0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font>
 
;<br/>
<a NAME="E7:109"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">s0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</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="c5">x</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="c4">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">s0</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_2"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<font color="Maroon" title="c5">x</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="c4">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">s0</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:109_2"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">s0</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:109_2"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">s0</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x9</font> = <font color="Maroon" title="c5">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109_2"><span class="lab"><font color="Green" title="E44">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c6">n</font> as    <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">by</span> <span class="lab"><a class="txt" href="#E3:109_2"><span class="lab"><font color="Green" title="E45">A7</font></span></a></span>;<br/>
<a NAME="E7:109_2"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<font color="Maroon" title="c4">s1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</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="c4">s1</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>;<br/>
<font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">s0</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_2"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s0</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T14" title="SEQ_1:th.14">SEQ_1:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">q3</font></span>)</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">q3</font>

;<br/>
<a NAME="E9:109_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">x9</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">q3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E7:109_2"><span class="lab"><font color="Green" title="E47">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:109_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T82" target="_self" title="MEASURE6:th.82">Th20</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="E8:109"/>
<font color="Maroon" title="c4">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">s0</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></span>;<br/>
<a NAME="E9:109"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">s0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E7:109"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="rcomp_1.html#D4" title="RCOMP_1:def.4">RCOMP_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:109"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:109"><span class="lab"><font color="Green" title="E42">A4</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>
