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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  &amp; <font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">real-bounded</a>  implies  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:72"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 <span class="kw">and </span><br/><a NAME="E2:72"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">real-bounded</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">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">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rcomp_1.html#D4" onmouseover="rs('rcomp_1/D4')" onmouseout="rh()">RCOMP_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Maroon" title="c2">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">X</font> or  not <font color="Maroon" title="c2">s1</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  or  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:72"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E4:72"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c2">s1</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E5:72"/><span class="lab"><font color="Green" title="E35">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="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">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="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_1"/>
<font color="Maroon" title="c3">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="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:72_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c4">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="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:72_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a></span>;<br/>
<a NAME="E6:72_1"/>
<font color="Maroon" title="c2">s1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/>
<a NAME="E7:72_1"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:72_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="valued_1.html#T10" onmouseover="rs('valued_1/T10')" onmouseout="rh()">VALUED_1:10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:72"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
 not  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E37">A9</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s1</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:72_2"/>
 not <font color="Maroon" title="c2">s1</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:72_2"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c2">s1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T5" onmouseover="rs('seq_1/T5')" onmouseout="rh()">SEQ_1:5</a></span>;<br/><a NAME="E4:72_2"/>
<font color="Maroon" title="c3">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:72_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a>, <a class="txt" href="#E3:72_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E38">A11</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><a NAME="E1:72_3"/><span class="lab"><font color="Green" title="E39">A12</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="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">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="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> or <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_3_1"/>
<font color="Maroon" title="c3">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="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:72_3_1"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<font color="Maroon" title="c4">x</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="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:72_3_1"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_3_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a></span>;<br/>
<a NAME="E6:72_3_1"/>
<font color="Maroon" title="c2">s1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/>
<a NAME="E7:72_3_1"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:72_3_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_3_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="ref" href="valued_1.html#T10" onmouseover="rs('valued_1/T10')" onmouseout="rh()">VALUED_1:10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:72_3"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:72_3"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E7:72"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="measure6.html#T38" target="_self" onmouseover="rs('measure6/T38')" onmouseout="rh()">Th38</a></span>;<br/><a NAME="E4:72_3"/><span class="kw">then </span>
 not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">real-bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T39" target="_self" onmouseover="rs('measure6/T39')" onmouseout="rh()">Th39</a></span>;<br/><span class="kw">hence </span><a NAME="E5:72_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E1:72_3"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="xxreal_2.html#T45" onmouseover="rs('xxreal_2/T45')" onmouseout="rh()">XXREAL_2:45</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:72"/><span class="kw">then </span>
<font color="Maroon" title="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E7:72"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="seq_2.html#T21" onmouseover="rs('seq_2/T21')" onmouseout="rh()">SEQ_2:21</a></span>;<br/>
<a NAME="E10:72"/><span class="kw">then </span>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E35">A5</font></span></a></span>;<br/>
<a NAME="E11:72"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E7:72"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="seq_2.html#T22" onmouseover="rs('seq_2/T22')" onmouseout="rh()">SEQ_2:22</a></span>;<br/>
<a NAME="E12:72"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E13:72"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">Inv</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
