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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/>for <font color="Olive" title="b3">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/>for <font color="Olive" title="b4">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b4">C</font> holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">B</font>;<br/>
<a NAME="E1:53_1_1"/><i><font color="Green" title="E38">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp;  ex <font color="Olive" title="b1">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="ref" href="prob_4.html#D5" target="_self" title="PROB_4:def.5">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font>] )</span><br/>

<b>thus </b><a NAME="E4:53_1_1_1"/>
( <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font>] )
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="prob_4.html#T26" target="_self" title="PROB_4:th.26">Th26</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">comP</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>, <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <b> such that </b><br/><a NAME="E3:53_1_1"/><i><font color="Green" title="E39">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="prob_4.html#E1:53_1_1"><i><font color="Green" title="E38">A1</font></i></a>);<br/></i>
<a NAME="E4:53_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<b>consider </b><font color="Maroon" title="c6">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:53_1_1_2"/><i><font color="Green" title="E41">A5</font></i>: 
( <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp;  ex <font color="Olive" title="b1">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="ref" href="prob_4.html#D5" target="_self" title="PROB_4:def.5">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font> <b>by </b><i><a class="txt" href="prob_4.html#E2:53_1_1_2"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<a NAME="E4:53_1_1_2"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="prob_4.html#E2:53_1_1_2"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<a NAME="E5:53_1_1_2"/>
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/><font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> implies <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_2_1"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:53_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">A</font> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:53_1_1_2_1"/>
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D13" title="PROB_1:def.13">PROB_1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:53_1_1_2"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="prob_4.html#E4:53_1_1_2"><i><font color="Green" title="E42">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:53_1_1"/><i><font color="Green" title="E41">A7</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E1:53_1_1_3"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E2:53_1_1_3"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="prob_4.html#E4:53_1_1"><i><font color="Green" title="E40">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:53_1_1"/><i><font color="Green" title="E42">A8</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/>for <font color="Olive" title="b2">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> implies for <font color="Olive" title="b1">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_4"/><i><font color="Green" title="E43">A9</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font></span><br/>

<b>let </b><font color="Maroon" title="c6">C</font> be    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font></span><br/>

<a NAME="E2:53_1_1_4"/>
<font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_4"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="prob_4.html#D5" target="_self" title="PROB_4:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:53_1_1_4"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="prob_4.html#E1:53_1_1_4"><i><font color="Green" title="E43">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:53_1_1"/><i><font color="Green" title="E43">A10</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="prob_4.html#T25" target="_self" title="PROB_4:th.25">Th25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">C</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> <b>by </b><i><a class="ref" href="prob_4.html#T25" target="_self" title="PROB_4:th.25">Th25</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">B</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<a NAME="E9:53_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">C</font>
 
;<br/>
<b>then </b><i><font color="Green" title="E44">A11</font></i>: <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="prob_4.html#E6:53_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="prob_1.html#T42" title="PROB_1:th.42">PROB_1:42</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="prob_1.html#T64" title="PROB_1:th.64">PROB_1:64</a></i>
;<br/>
<a NAME="E11:53_1_1"/><i><font color="Green" title="E45">A12</font></i>: 
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">Omega</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_6"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">Omega</font> = 
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c1">Omega</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">Omega</font>
<b>by </b><i><a class="txt" href="prob_4.html#E6:53_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="prob_4.html#E7:53_1_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="prob_1.html#T43" title="PROB_1:th.43">PROB_1:43</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="prob_1.html#D13" title="PROB_1:def.13">PROB_1:def 13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:53_1_1"/><i><font color="Green" title="E46">A13</font></i>: 
for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">BSeq</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> holds <br/> <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b1">BSeq</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">BSeq</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">BSeq</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> implies  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_7"/><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c7">BSeq</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c7">BSeq</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Omega</font> ;<br/>
<a NAME="E3:53_1_1_7"/><i><font color="Green" title="E48">A15</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">BSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:53_1_1_7"/>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_7"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">CSeq</font> being    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><b> such that </b><br/><a NAME="E7:53_1_1_7"/><i><font color="Green" title="E49">A16</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K6" title="PROB_4:func.6">ProbPart</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_4.html#T32" target="_self" title="PROB_4:th.32">Th32</a></i>;<br/>
<a NAME="E8:53_1_1_7"/><i><font color="Green" title="E50">A17</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> )</span><br/>

<a NAME="E1:53_1_1_7_1"/>
<font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K6" title="PROB_4:func.6">ProbPart</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E7:53_1_1_7"><i><font color="Green" title="E49">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:53_1_1_7_1"/>
( <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="ref" href="prob_4.html#D7" target="_self" title="PROB_4:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c11">DSeq</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font><b> such that </b><br/><a NAME="E10:53_1_1_7"/><i><font color="Green" title="E51">A18</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_4.html#T33" target="_self" title="PROB_4:th.33">Th33</a></i>;<br/>
<a NAME="E11:53_1_1_7"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font></span><br/>

<a NAME="E1:53_1_1_7_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E8:53_1_1_7"><i><font color="Green" title="E50">A17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:53_1_1_7_2"/>
<font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c12">n</font> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E10:53_1_1_7"><i><font color="Green" title="E51">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c12">ESeq</font> being    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><b> such that </b><br/><a NAME="E13:53_1_1_7"/><i><font color="Green" title="E52">A19</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">ESeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ESeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="prob_4.html#T34" target="_self" title="PROB_4:th.34">Th34</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:53_1_1_7"/><i><font color="Green" title="E53">A20</font></i>: 
<font color="Maroon" title="c13">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">CSeq</font></span>)</span>
 ;<br/>
<a NAME="E16:53_1_1_7"/>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c10">CSeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T46" title="PROB_1:th.46">PROB_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">B</font> = <font color="Maroon" title="c13">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font> <b>by </b><i><a class="txt" href="prob_4.html#E15:53_1_1_7"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/>
<a NAME="E18:53_1_1_7"/><i><font color="Green" title="E54">A21</font></i>: 
<font color="Maroon" title="c14">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_7_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c16">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_7_3"/><i><font color="Green" title="E55">A22</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z</font> &amp; <font color="Maroon" title="c16">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">CSeq</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E15:53_1_1_7"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:53_1_1_7_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">CSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:53_1_1_7_3"/><i><font color="Green" title="E56">A23</font></i>: 
( <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c16">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_7_3"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">n</font> = <font color="Maroon" title="c17">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prob_4.html#E6:53_1_1_7_3"><i><font color="Green" title="E56">A23</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">P</font> = <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font>;<br/>
<a NAME="E8:53_1_1_7_3"/><i><font color="Green" title="E57">A24</font></i>: 
<font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E8:53_1_1_7"><i><font color="Green" title="E50">A17</font></i></a></i>;<br/>
<a NAME="E9:53_1_1_7_3"/>
 ex <font color="Olive" title="b1">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> )</span><br/>

<b>thus </b><a NAME="E1:53_1_1_7_3_1"/>
( <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_7"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="prob_4.html#E3:53_1_1_7_3"><i><font color="Green" title="E55">A22</font></i></a>, <a class="txt" href="prob_4.html#E6:53_1_1_7_3"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="prob_4.html#E8:53_1_1_7_3"><i><font color="Green" title="E57">A24</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:53_1_1_7_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E19:53_1_1_7"/><i><font color="Green" title="E55">A25</font></i>: 
 ex <font color="Olive" title="b1">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> st  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c15">C</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c14">B</font>;<br/>
<i><font color="Green" title="E56">A26</font></i>:  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c14">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>
<br/>.= 
<font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p3">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c14">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E18:53_1_1_7"><i><font color="Green" title="E54">A21</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c16">C</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c14">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font> ;<br/>
<a NAME="E3:53_1_1_7_4"/>
<font color="Maroon" title="c16">C</font> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:53_1_1_7_4_2"/><i><font color="Green" title="E57">A27</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">C</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_7_4_2_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font></span>)</span></span><br/>

<a NAME="E2:53_1_1_7_4_2_1"/><b>then </b><i><font color="Green" title="E58">A28</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> &amp; not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">CSeq</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="prob_4.html#E15:53_1_1_7"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:53_1_1_7_4_2_1"/><i><font color="Green" title="E59">A29</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Z</font> &amp; <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:53_1_1_7_4_2_1"/><i><font color="Green" title="E60">A30</font></i>: 
( <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_7"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="prob_4.html#E4:53_1_1_7_4_2_1"><i><font color="Green" title="E59">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prob_4.html#E6:53_1_1_7_4_2_1"><i><font color="Green" title="E60">A30</font></i></a></i>;<br/>
<a NAME="E8:53_1_1_7_4_2_1"/>
not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_4_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_1_7_4_2_1_1"/><i><font color="Green" title="E61">A31</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:53_1_1_7_4_2_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">CSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:53_1_1_7_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">CSeq</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:53_1_1_7_4_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="prob_4.html#E2:53_1_1_7_4_2_1"><i><font color="Green" title="E58">A28</font></i></a>, <a class="txt" href="prob_4.html#E1:53_1_1_7_4_2_1_1"><i><font color="Green" title="E61">A31</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:53_1_1_7_4_2_1"/><b>then </b><i><font color="Green" title="E61">A32</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E4:53_1_1_7_4_2_1"><i><font color="Green" title="E59">A29</font></i></a>, <a class="txt" href="prob_4.html#E6:53_1_1_7_4_2_1"><i><font color="Green" title="E60">A30</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:53_1_1_7_4_2_1"/>
 ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_4_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:53_1_1_7_4_2_1_2"/><i><font color="Green" title="E62">A33</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">DSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c20">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font> )</span><br/>

<b>thus </b><a NAME="E2:53_1_1_7_4_2_1_2"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E10:53_1_1_7"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="prob_4.html#E9:53_1_1_7_4_2_1"><i><font color="Green" title="E61">A32</font></i></a>, <a class="txt" href="prob_4.html#E1:53_1_1_7_4_2_1_2"><i><font color="Green" title="E62">A33</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:53_1_1_7_4_2_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:53_1_1_7_4_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_4_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font></span>)</span> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_7_4_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c18">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_7_4_2_2"/><i><font color="Green" title="E58">A34</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Z</font> &amp; <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">DSeq</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:53_1_1_7_4_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">DSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:53_1_1_7_4_2_2"/><i><font color="Green" title="E59">A35</font></i>: 
( <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">DSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_7_4_2_2"><i><font color="Green" title="E58">A34</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prob_4.html#E6:53_1_1_7_4_2_2"><i><font color="Green" title="E59">A35</font></i></a></i>;<br/>
<a NAME="E8:53_1_1_7_4_2_2"/><i><font color="Green" title="E60">A36</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c11">DSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">ESeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font> )
 
<b>by </b><i><a class="txt" href="prob_4.html#E13:53_1_1_7"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="prob_4.html#E3:53_1_1_7_4_2_2"><i><font color="Green" title="E58">A34</font></i></a>, <a class="txt" href="prob_4.html#E6:53_1_1_7_4_2_2"><i><font color="Green" title="E59">A35</font></i></a></i>;<br/>
<a NAME="E9:53_1_1_7_4_2_2"/>
<font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">ESeq</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E6:53_1_1_7_4_2_2"><i><font color="Green" title="E59">A35</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:53_1_1_7_4_2_2"/><b>then </b>
<font color="Maroon" title="c12">ESeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:53_1_1_7_4_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_4.html#E8:53_1_1_7_4_2_2"><i><font color="Green" title="E60">A36</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:53_1_1_7_4_2"/><b>then </b><i><font color="Green" title="E58">A37</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_7_4_2"><i><font color="Green" title="E57">A27</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:53_1_1_7_4_2"/>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c12">ESeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T46" title="PROB_1:th.46">PROB_1:46</a></i>;<br/>
<a NAME="E5:53_1_1_7_4_2"/><b>then </b><i><font color="Green" title="E59">A38</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/>
<a NAME="E6:53_1_1_7_4_2"/>
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_4_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:53_1_1_7_4_2_3"/>
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font> holds <br/><font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_4_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font> implies <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_7_4_2_3_1"/><i><font color="Green" title="E60">A39</font></i>: 
<font color="Maroon" title="c17">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:53_1_1_7_4_2_3_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">ESeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:53_1_1_7_4_2_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c17">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">ESeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_7_4_2_3_1"><i><font color="Green" title="E60">A39</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:53_1_1_7_4_2_3_1"/>
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="prob_4.html#E13:53_1_1_7"><i><font color="Green" title="E52">A19</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_1_1_7_4_2_3"/>
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">ESeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="prob_4.html#T9" target="_self" title="PROB_4:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:53_1_1_7_4_2"/>
<font color="Maroon" title="c16">C</font> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_7_4_2"><i><font color="Green" title="E58">A37</font></i></a>, <a class="txt" href="prob_4.html#E5:53_1_1_7_4_2"><i><font color="Green" title="E59">A38</font></i></a>, <a class="ref" href="prob_4.html#D4" target="_self" title="PROB_4:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c17">C</font> being    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font><b> such that </b><br/><a NAME="E5:53_1_1_7_4"/><i><font color="Green" title="E57">A40</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c17">C</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_7_4"><i><font color="Green" title="E56">A26</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c17">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c17">C</font></span><br/>

<b>thus </b><a NAME="E6:53_1_1_7_4"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c17">C</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E5:53_1_1_7_4"><i><font color="Green" title="E57">A40</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E20:53_1_1_7"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">n</font>, <font color="Maroon" title="c16">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">m</font> implies <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_7_5"/>
<font color="Maroon" title="c15">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font></span><br/>

<a NAME="E2:53_1_1_7_5"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_7"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></i>;<br/>
<a NAME="E3:53_1_1_7_5"/><b>then </b><i><font color="Green" title="E56">A41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<i><font color="Green" title="E57">A42</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">F</font> = 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">CSeq</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E5:53_1_1_7_5"/>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_7_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_7_5_2_1"/>
( <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">F</font> or  not <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_7_5_2_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_7_5_2_1_1"/>
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font></span><br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_7_5_2_1_1"/>
<font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E8:53_1_1_7"><i><font color="Green" title="E50">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_7_5_2_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_7_5_2_1_2"/><i><font color="Green" title="E58">A43</font></i>: 
not <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font></span><br/><div class="add"><b>then </b><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> = 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font>
<b>by </b><i><a class="txt" href="prob_4.html#E4:53_1_1_7_5"><i><font color="Green" title="E57">A42</font></i></a>, <a class="txt" href="prob_4.html#E1:53_1_1_7_5_2_1_2"><i><font color="Green" title="E58">A43</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:53_1_1_7_5_2_1_2"/>
<font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:53_1_1_7_5"/><b>then </b>
( <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font> )
 
;<br/>
<a NAME="E7:53_1_1_7_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_7_5"><i><font color="Green" title="E56">A41</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<b>hence </b><a NAME="E8:53_1_1_7_5"/>
<font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">CSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E21:53_1_1_7"/><b>then </b><i><font color="Green" title="E56">A44</font></i>: 
<font color="Maroon" title="c10">CSeq</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>
 
<b>by </b><i><a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></i>;<br/>
<a NAME="E22:53_1_1_7"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c9">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_6"><b>proof </b></a><div class="add">

<a NAME="E1:53_1_1_7_6"/><i><font color="Green" title="E57">A45</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span><br/>

<a NAME="E1:53_1_1_7_6_1"/><i><font color="Green" title="E58">A46</font></i>: 
( <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> is    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="prob_4.html#E8:53_1_1_7"><i><font color="Green" title="E50">A17</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">C</font> being    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font><b> such that </b><br/><a NAME="E3:53_1_1_7_6_1"/><i><font color="Green" title="E59">A47</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 ;<br/>
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">C</font>
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_7_6_1"><i><font color="Green" title="E58">A46</font></i></a>, <a class="txt" href="prob_4.html#E3:53_1_1_7_6_1"><i><font color="Green" title="E59">A47</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E5:53_1_1_7_6_1"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_4.html#E6:53_1_1"><i><font color="Green" title="E42">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:53_1_1_7_6"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">CSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_7_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">CSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_7_6_2"/><i><font color="Green" title="E58">A48</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">CSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c15">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_7_6_2"><i><font color="Green" title="E58">A48</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> = 
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_7_6"><i><font color="Green" title="E57">A45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">CSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>
<b>hence </b><a NAME="E4:53_1_1_7_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">CSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:53_1_1_7_6"/><b>then </b><i><font color="Green" title="E58">A49</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">CSeq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">C1</font> being    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font><b> such that </b><br/><a NAME="E5:53_1_1_7_6"/><i><font color="Green" title="E59">A50</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">C1</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E19:53_1_1_7"><i><font color="Green" title="E55">A25</font></i></a></i>;<br/>
<a NAME="E6:53_1_1_7_6"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">CSeq</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E6:53_1_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="prob_4.html#E15:53_1_1_7"><i><font color="Green" title="E53">A20</font></i></a>, <a class="txt" href="prob_4.html#E5:53_1_1_7_6"><i><font color="Green" title="E59">A50</font></i></a></i>;<br/>
<b>then </b><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c9">F</font></span>)</span> = 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">CSeq</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c10">CSeq</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E8:53_1_1_7_6"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c9">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_4.html#E21:53_1_1_7"><i><font color="Green" title="E56">A44</font></i></a>, <a class="txt" href="prob_4.html#E3:53_1_1_7_6"><i><font color="Green" title="E58">A49</font></i></a>, <a class="ref" href="prob_3.html#T51" title="PROB_3:th.51">PROB_3:51</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:53_1_1_7"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:53_1_1"/><i><font color="Green" title="E47">A51</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">B</font> implies <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_8"/><i><font color="Green" title="E48">A52</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c9">BSeq</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font><b> such that </b><br/><a NAME="E3:53_1_1_8"/><i><font color="Green" title="E49">A53</font></i>: 
( <font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 holds <br/><font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <b>by </b><i><a class="ref" href="prob_4.html#T3" target="_self" title="PROB_4:th.3">Th3</a></i>;<br/>
<a NAME="E4:53_1_1_8"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_8_1_1"/>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_8_1_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font></span><br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_8_1_1_1"/>
<font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_8"><i><font color="Green" title="E49">A53</font></i></a>, <a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_8_1_1_2"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font></span><br/><div class="add"><a NAME="E2:53_1_1_8_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:53_1_1_8_1_1_2"/><b>then </b><i><font color="Green" title="E50">A54</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_8_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_8_1_1_2_1"/>
( <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_8_1_1_2"><i><font color="Green" title="E50">A54</font></i></a>, <a class="ref" href="nat_1.html#T22" title="NAT_1:th.22">NAT_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_8_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_8_1_1_2_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font></span><br/><div class="add"><a NAME="E2:53_1_1_8_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_8"><i><font color="Green" title="E49">A53</font></i></a></i>;<br/><b>hence </b><a NAME="E3:53_1_1_8_1_1_2_1_1"/>
<font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T42" title="PROB_1:th.42">PROB_1:42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_8_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_8_1_1_2_1_2"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font></span><br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_8_1_1_2_1_2"/>
<font color="Maroon" title="c9">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_8"><i><font color="Green" title="E49">A53</font></i></a>, <a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">BSeq</font> = <font color="Maroon" title="c9">BSeq</font> as    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/>
<a NAME="E6:53_1_1_8"/><i><font color="Green" title="E50">A55</font></i>: 
( <font color="Maroon" title="c10">BSeq</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> &amp;  <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c10">BSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c8">B</font> )
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_8"><i><font color="Green" title="E48">A52</font></i></a>, <a class="txt" href="prob_4.html#E3:53_1_1_8"><i><font color="Green" title="E49">A53</font></i></a>, <a class="ref" href="prob_4.html#T4" target="_self" title="PROB_4:th.4">Th4</a></i>;<br/>
<a NAME="E7:53_1_1_8"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_8_2"><b>proof </b></a><div class="add">

<i><font color="Green" title="E51">A56</font></i>: <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_8"><i><font color="Green" title="E49">A53</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E2:53_1_1_8_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<b>then </b><i><font color="Green" title="E52">A57</font></i>: <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_8"><i><font color="Green" title="E49">A53</font></i></a>, <a class="txt" href="prob_4.html#E1:53_1_1_8_2"><i><font color="Green" title="E51">A56</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E4:53_1_1_8_2"/><i><font color="Green" title="E53">A58</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_8_2_3"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span>;<br/>
<a NAME="E1:53_1_1_8_2_3"/><i><font color="Green" title="E54">A59</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_8_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:53_1_1_8_2_3_1"/>
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_8_2"><i><font color="Green" title="E52">A57</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_8"><i><font color="Green" title="E49">A53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E10:53_1_1"><i><font color="Green" title="E44">A11</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:53_1_1_8_2_3"/><i><font color="Green" title="E55">A60</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_8_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_8_2_3_2"/><i><font color="Green" title="E56">A61</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:53_1_1_8_2_3_2"/><i><font color="Green" title="E57">A62</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_8_2_3_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:53_1_1_8_2_3_2_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E2:53_1_1_8_2_3_2_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_8_2_3_2"><i><font color="Green" title="E56">A61</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_8"><i><font color="Green" title="E49">A53</font></i></a>, <a class="txt" href="prob_4.html#E2:53_1_1_8_2_3_2"><i><font color="Green" title="E57">A62</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E10:53_1_1"><i><font color="Green" title="E44">A11</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:53_1_1_8_2_3"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="prob_4.html#E1:53_1_1_8_2_3"><i><font color="Green" title="E54">A59</font></i></a>, <a class="txt" href="prob_4.html#E2:53_1_1_8_2_3"><i><font color="Green" title="E55">A60</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c11">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> implies <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E5:53_1_1_8_2"/><i><font color="Green" title="E54">A63</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c12">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:53_1_1_8_2"/><i><font color="Green" title="E55">A64</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E5:53_1_1_8_2"><i><font color="Green" title="E54">A63</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<a NAME="E8:53_1_1_8_2"/>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E9:53_1_1_8_2"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_4.html#E4:53_1_1_8_2"><i><font color="Green" title="E53">A58</font></i></a>, <a class="txt" href="prob_4.html#E7:53_1_1_8_2"><i><font color="Green" title="E55">A64</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:53_1_1_8"/><b>then </b>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c10">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#T3" title="PROB_1:th.3">PROB_1:3</a></i>;<br/>
<b>hence </b><a NAME="E9:53_1_1_8"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_4.html#E12:53_1_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="prob_4.html#E6:53_1_1_8"><i><font color="Green" title="E50">A55</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:53_1_1"/><i><font color="Green" title="E48">A65</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font> implies <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_9"/><i><font color="Green" title="E49">A66</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E13:53_1_1"><i><font color="Green" title="E47">A51</font></i></a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>
<br/>.= 
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font>
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_9"><i><font color="Green" title="E49">A66</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>
;<br/>
<b>hence </b><a NAME="E3:53_1_1_9"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:53_1_1"/><i><font color="Green" title="E49">A67</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font> implies <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_10"/>
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span><br/>

<a NAME="E2:53_1_1_10"/><b>then </b>
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E14:53_1_1"><i><font color="Green" title="E48">A65</font></i></a></i>;<br/>
<a NAME="E3:53_1_1_10"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E5:53_1_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>
<a NAME="E4:53_1_1_10"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_1_10"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:53_1_1"/><i><font color="Green" title="E50">A68</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">A</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c1">Omega</font> as    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> <b>by </b><i><a class="ref" href="prob_1.html#T53" title="PROB_1:th.53">PROB_1:53</a></i>;<br/>
<a NAME="E2:53_1_1_11"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E15:53_1_1"><i><font color="Green" title="E49">A67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:53_1_1_11"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="prob_4.html#E11:53_1_1"><i><font color="Green" title="E45">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E17:53_1_1"/><i><font color="Green" title="E51">A69</font></i>: 
for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">BSeq</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>



<a NAME="E1:53_1_1_12"/>
<font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c8">n</font> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E2:53_1_1_12"/>
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="prob_4.html#E16:53_1_1"><i><font color="Green" title="E50">A68</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E18:53_1_1"/><i><font color="Green" title="E52">A70</font></i>: 
for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">BSeq</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>



<a NAME="E1:53_1_1_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E2:53_1_1_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="prob_4.html#E17:53_1_1"><i><font color="Green" title="E51">A69</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E19:53_1_1"/><i><font color="Green" title="E53">A71</font></i>: 
for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">BSeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> holds <br/><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">BSeq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">BSeq</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">BSeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> implies <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_14"/><i><font color="Green" title="E54">A72</font></i>: 
<font color="Maroon" title="c7">BSeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a></span><br/>

<a NAME="E2:53_1_1_14"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font>, <font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font> )</span><br/>



<b>assume </b><a NAME="E1:53_1_1_14_1"/><i><font color="Green" title="E55">A73</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font></span><br/>

<a NAME="E2:53_1_1_14_1"/><i><font color="Green" title="E56">A74</font></i>: 
( <font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c8">n</font> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">m</font> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<a NAME="E3:53_1_1_14_1"/>
<font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_14"><i><font color="Green" title="E54">A72</font></i></a>, <a class="txt" href="prob_4.html#E1:53_1_1_14_1"><i><font color="Green" title="E55">A73</font></i></a>, <a class="ref" href="prob_1.html#D7" title="PROB_1:def.7">PROB_1:def 7</a></i>;<br/>
<a NAME="E4:53_1_1_14_1"/><b>then </b>
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E15:53_1_1"><i><font color="Green" title="E49">A67</font></i></a>, <a class="txt" href="prob_4.html#E2:53_1_1_14_1"><i><font color="Green" title="E56">A74</font></i></a></i>;<br/>
<a NAME="E5:53_1_1_14_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E6:53_1_1_14_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:53_1_1_14"/>
<font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T12" title="SEQM_3:th.12">SEQM_3:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E20:53_1_1"/><i><font color="Green" title="E54">A75</font></i>: 
for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">BSeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> holds <br/>( <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">BSeq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">BSeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">BSeq</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">BSeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> implies ( <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> ) )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_15"/><i><font color="Green" title="E55">A76</font></i>: 
<font color="Maroon" title="c7">BSeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )</span><br/>

<a NAME="E2:53_1_1_15"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2</span><br/>

<a NAME="E1:53_1_1_15_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prob_4.html#E18:53_1_1"><i><font color="Green" title="E52">A70</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E2:53_1_1_15_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:53_1_1_15"/><b>then </b>
<font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:53_1_1_15"/>
( <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 <b>by </b><i><a class="txt" href="prob_4.html#E19:53_1_1"><i><font color="Green" title="E53">A71</font></i></a>, <a class="txt" href="prob_4.html#E1:53_1_1_15"><i><font color="Green" title="E55">A76</font></i></a>, <a class="ref" href="seq_4.html#T51" title="SEQ_4:th.51">SEQ_4:51</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E21:53_1_1"/>
for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>  st <font color="Olive" title="b1">BSeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> holds <br/>( <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">BSeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b1">BSeq</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">BSeq</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">BSeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> implies ( <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_16"/><i><font color="Green" title="E55">A77</font></i>: 
<font color="Maroon" title="c7">BSeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> )</span><br/>

<b>set </b><font color="Maroon" title="c8">CSeq</font> =  <a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font>;<br/>
<i><font color="Green" title="E56">A78</font></i>: <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> = 
<font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#T40" title="PROB_3:th.40">PROB_3:40</a></i>
<br/>.= 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E12:53_1_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="prob_3.html#T41" title="PROB_3:th.41">PROB_3:41</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E3:53_1_1_16"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_16_2"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<a NAME="E1:53_1_1_16_2"/><i><font color="Green" title="E57">A79</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_16_2_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_16"><i><font color="Green" title="E55">A77</font></i></a>, <a class="ref" href="prob_4.html#T21" target="_self" title="PROB_4:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:53_1_1_16_2"/><i><font color="Green" title="E58">A80</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_16_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:53_1_1_16_2_2"/><i><font color="Green" title="E59">A81</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:53_1_1_16_2_2"/><i><font color="Green" title="E60">A82</font></i>: 
( <font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">k</font> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<a NAME="E3:53_1_1_16_2_2"/><i><font color="Green" title="E61">A83</font></i>: 
<font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_16"><i><font color="Green" title="E55">A77</font></i></a>, <a class="ref" href="prob_2.html#T15" title="PROB_2:th.15">PROB_2:15</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_16_2_2"><i><font color="Green" title="E59">A81</font></i></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">@Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E13:53_1_1"><i><font color="Green" title="E47">A51</font></i></a>, <a class="txt" href="prob_4.html#E1:53_1_1_16"><i><font color="Green" title="E55">A77</font></i></a>, <a class="txt" href="prob_4.html#E2:53_1_1_16_2_2"><i><font color="Green" title="E60">A82</font></i></a>, <a class="ref" href="prob_4.html#T22" target="_self" title="PROB_4:th.22">Th22</a></i>
<br/>.= 
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E1:53_1_1_16"><i><font color="Green" title="E55">A77</font></i></a>, <a class="ref" href="prob_4.html#T21" target="_self" title="PROB_4:th.21">Th21</a></i>
<br/>.= 
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c4">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_4.html#E3:53_1_1_16_2_2"><i><font color="Green" title="E61">A83</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:53_1_1_16_2"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="prob_4.html#E1:53_1_1_16_2"><i><font color="Green" title="E57">A79</font></i></a>, <a class="txt" href="prob_4.html#E2:53_1_1_16_2"><i><font color="Green" title="E58">A80</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:53_1_1_16"/>
( <font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comP</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prob_4.html#E20:53_1_1"><i><font color="Green" title="E54">A75</font></i></a>, <a class="txt" href="prob_4.html#E1:53_1_1_16"><i><font color="Green" title="E55">A77</font></i></a>, <a class="txt" href="prob_4.html#E2:53_1_1_16"><i><font color="Green" title="E56">A78</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">comP</font> = <font color="Maroon" title="c4">comP</font> as    <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> <b>by </b><i><a class="txt" href="prob_4.html#E5:53_1_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="prob_4.html#E11:53_1_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="prob_4.html#E13:53_1_1"><i><font color="Green" title="E47">A51</font></i></a>, <a class="ref" href="prob_2.html#T20" title="PROB_2:th.20">PROB_2:20</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">comP</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/>for <font color="Olive" title="b2">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c7">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font></span><br/>

<b>thus </b><a NAME="E23:53_1_1"/>
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/>for <font color="Olive" title="b2">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c7">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E6:53_1_1"><i><font color="Green" title="E42">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
