<?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#M1" title="HIDDEN:mode.1">set</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">F</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">F</font> is  <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a>  holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</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">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c2">F</font> is  <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a>  holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c2">F</font> is  <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a>  implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">PF</font> =  <a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font>;<br/>
<span class="kw">assume </span><a NAME="E1:34"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:34_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:34_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E5:34_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c5">z</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E6:34_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</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"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E7:34_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:34_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E10:34_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="mesfunc9.html#T25" target="_self" title="MESFUNC9:th.25">Th25</a></span>;<br/><a NAME="E11:34_1"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c5">z</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">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E12:34_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</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"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E13:34_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <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/><a NAME="E14:34_1"/>
<font color="Maroon" title="c5">z</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">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E9:34_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="mesfunc9.html#T22" target="_self" title="MESFUNC9:th.22">Th22</a></span>;<br/><a NAME="E15:34_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</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="#E11:34_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E16:34_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E13:34_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="txt" href="#E10:34_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="mesfunc9.html#D5" target="_self" title="MESFUNC9:def.5">Def5</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="E3:34"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:34"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:34_2"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:34_2"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E5:34_2"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c5">z</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E6:34_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</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"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_2"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E7:34_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:34_2"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E10:34_2"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="mesfunc9.html#T23" target="_self" title="MESFUNC9:th.23">Th23</a></span>;<br/><a NAME="E11:34_2"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c5">z</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">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E12:34_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</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"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/><a NAME="E13:34_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <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/><a NAME="E14:34_2"/>
<font color="Maroon" title="c5">z</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">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E9:34_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="mesfunc9.html#T22" target="_self" title="MESFUNC9:th.22">Th22</a></span>;<br/><a NAME="E15:34_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</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="#E11:34_2"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E16:34_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E13:34_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E10:34_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="mesfunc9.html#D5" target="_self" title="MESFUNC9:def.5">Def5</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="E6:34"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:34"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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>
