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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">M</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  holds <br/> ex <font color="Olive" title="b3">F</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b3">F</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <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">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">M</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  holds <br/> ex <font color="Olive" title="b2">F</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b2">F</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</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"> ( <font color="Maroon" title="c3">M</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  implies  ex <font color="Olive" title="b1">F</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:109"/>
<font color="Maroon" title="c3">M</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</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">  ex <font color="Olive" title="b1">F</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">E</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E3:109"/><span class="lab"><font color="Green" title="E76">A1</font></span>: 
( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) &amp;  <a href="measur11.html#K3" title="MEASUR11:func.3">Union</a> <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> $1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font>;<br/>
<a NAME="E4:109"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.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="prob_3.html#D2" onmouseover="rs('prob_3/D2')" onmouseout="rh()">PROB_3:def 2</a></span>;<br/>
<a NAME="E5:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" onmouseover="rs('measure8/D2')" onmouseout="rh()">MEASURE8:def 2</a></span>;<br/>
<a NAME="E6:109"/><span class="lab"><font color="Green" title="E78">A3</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="109_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_1"/><span class="lab"><font color="Green" title="E79">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:109_1"/><span class="lab"><font color="Green" title="E80">A5</font></span>: 
<font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" onmouseover="rs('measure8/D2')" onmouseout="rh()">MEASURE8:def 2</a></span>;<br/>
<a NAME="E3:109_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#D2" onmouseover="rs('prob_3/D2')" onmouseout="rh()">PROB_3:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:109_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109_1"><span class="lab"><font color="Green" title="E79">A4</font></span></a>, <a class="txt" href="#E2:109_1"><span class="lab"><font color="Green" title="E80">A5</font></span></a>, <a class="ref" href="finsub_1.html#D1" onmouseover="rs('finsub_1/D1')" onmouseout="rh()">FINSUB_1:def 1</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="E7:109"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E5:109"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="txt" href="#E6:109"><span class="lab"><font color="Green" title="E78">A3</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">F</font> =  <a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">E</font> as    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" onmouseover="rs('measure8/D2')" onmouseout="rh()">MEASURE8:def 2</a></span>;<br/>
<a NAME="E9:109"/><span class="lab"><font color="Green" title="E79">A6</font></span>: 
<font color="Maroon" title="c5">F</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T11" onmouseover="rs('prob_3/T11')" onmouseout="rh()">PROB_3:11</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> $1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> ;<br/>
<a NAME="E10:109"/>
<font color="Maroon" title="c5">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.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="prob_3.html#D2" onmouseover="rs('prob_3/D2')" onmouseout="rh()">PROB_3:def 2</a></span>;<br/>
<a NAME="E11:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A7</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109"><span class="lab"><font color="Green" title="E76">A1</font></span></a></span>;<br/>
<a NAME="E12:109"/><span class="lab"><font color="Green" title="E81">A8</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c6">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:109_2"/><span class="lab"><font color="Green" title="E82">A9</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</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"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:109_2"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109"><span class="lab"><font color="Green" title="E76">A1</font></span></a></span>;<br/>
<a NAME="E3:109_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109_2"><span class="lab"><font color="Green" title="E82">A9</font></span></a>, <a class="ref" href="xxreal_3.html#T16" onmouseover="rs('xxreal_3/T16')" onmouseout="rh()">XXREAL_3:16</a>, <a class="ref" href="xxreal_0.html#T4" onmouseover="rs('xxreal_0/T4')" onmouseout="rh()">XXREAL_0:4</a></span>;<br/>
<a NAME="E4:109_2"/><span class="lab"><font color="Green" title="E84">A11</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#D2" onmouseover="rs('prob_3/D2')" onmouseout="rh()">PROB_3:def 2</a></span>;<br/>
<a NAME="E5:109_2"/>
( <font color="Maroon" title="c5">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" onmouseover="rs('measure8/D2')" onmouseout="rh()">MEASURE8:def 2</a></span>;<br/>
<a NAME="E6:109_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_2"><span class="lab"><font color="Green" title="E84">A11</font></span></a>, <a class="ref" href="measure1.html#T33" onmouseover="rs('measure1/T33')" onmouseout="rh()">MEASURE1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:109_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_2"><span class="lab"><font color="Green" title="E83">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="E13:109"/><span class="lab"><font color="Green" title="E82">A12</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E11:109"><span class="lab"><font color="Green" title="E80">A7</font></span></a>, <a class="txt" href="#E12:109"><span class="lab"><font color="Green" title="E81">A8</font></span></a>);<br/></span>
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c5">F</font> = 
 <a href="measur11.html#K3" title="MEASUR11:func.3">Union</a> <font color="Maroon" title="c5">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:109"><span class="lab"><font color="Green" title="E79">A6</font></span></a>, <a class="ref" href="setlim_1.html#T63" onmouseover="rs('setlim_1/T63')" onmouseout="rh()">SETLIM_1:63</a></span>
<br/>.= 
 <a href="measur11.html#K3" title="MEASUR11:func.3">Union</a> <font color="Maroon" title="c4">E</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T15" onmouseover="rs('prob_3/T15')" onmouseout="rh()">PROB_3:15</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:109"/>
 ex <font color="Olive" title="b1">F</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109"><span class="lab"><font color="Green" title="E76">A1</font></span></a>, <a class="txt" href="#E9:109"><span class="lab"><font color="Green" title="E79">A6</font></span></a>, <a class="txt" href="#E13:109"><span class="lab"><font color="Green" title="E82">A12</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>
