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<span class="kw">reconsider </span><font color="Maroon" title="c4">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#T24" target="_self" onmouseover="rs('prob_4/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">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#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">B</font> <a href="tarski.html#R2" title="TARSKI: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/>
<span class="kw">set </span><font color="Maroon" title="c5">B</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<a NAME="E2:47_1_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#T24" target="_self" onmouseover="rs('prob_4/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<a NAME="E3:47_1_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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#K4" title="PROB_4:func.4">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">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">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">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:47_1_1_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
( <font color="Maroon" title="c7">B</font> <a href="tarski.html#R2" title="TARSKI: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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#D5" target="_self" onmouseover="rs('prob_4/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">B</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="c7">B</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E4:47_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="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="c7">B</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="prob_4.html#T25" target="_self" onmouseover="rs('prob_4/T25')" onmouseout="rh()">Th25</a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">comP</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E5:47_1_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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#K4" title="PROB_4:func.4">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="c6">comP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E3:47_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>);<br/></span>
<a NAME="E6:47_1_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 for <font color="Olive" title="b1">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI: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="c6">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">B</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">B</font> <a href="tarski.html#R2" title="TARSKI: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="c6">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">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="c7">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_2"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="c6">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">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="c7">B</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">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="c7">B</font></span><br/>

<a NAME="E2:47_1_1_2"/>
<font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_2"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="prob_4.html#D5" target="_self" onmouseover="rs('prob_4/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:47_1_1_2"/>
<font color="Maroon" title="c6">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">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">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="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E1:47_1_1_2"><span class="lab"><font color="Green" title="E38">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:47_1_1"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
 for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">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">V128</a>() holds <br/> <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b1">BSeq</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">BSeq</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">BSeq</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">V128</a>() implies  <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_3"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c7">BSeq</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">V128</a>()
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c7">BSeq</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Omega</font></span>)</span> ;<br/>
<a NAME="E3:47_1_1_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">F</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" onmouseover="rs('funct_2/T6')" onmouseout="rh()">FUNCT_2:6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">CSeq</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><span class="kw"> such that </span><br/><a NAME="E6:47_1_1_3"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  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="tarski.html#R2" title="TARSKI: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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#T31" target="_self" onmouseover="rs('prob_4/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E8:47_1_1_3"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<font color="Maroon" title="c11">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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c10">CSeq</font></span>)</span>
 ;<br/>
<a NAME="E9:47_1_1_3"/>
 <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c10">CSeq</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T17" onmouseover="rs('prob_1/T17')" onmouseout="rh()">PROB_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">B</font> = <font color="Maroon" title="c11">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_1_1_3"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">DSeq</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font><span class="kw"> such that </span><br/><a NAME="E12:47_1_1_3"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4: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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#T32" target="_self" onmouseover="rs('prob_4/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<a NAME="E13:47_1_1_3"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI: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="c14">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="c14">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="c14">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="c14">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="E1:47_1_1_3_1"/>
<font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI: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="c14">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1_1_3"><span class="lab"><font color="Green" title="E40">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:47_1_1_3_1"/>
( <font color="Maroon" title="c10">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI: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="c14">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="c14">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="c14">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="c14">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 class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#D7" target="_self" onmouseover="rs('prob_4/D7')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:47_1_1_3"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4: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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c14">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="E1:47_1_1_3_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="c14">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="c14">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1_1_3"><span class="lab"><font color="Green" title="E43">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:47_1_1_3_2"/>
<font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c14">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47_1_1_3"><span class="lab"><font color="Green" title="E42">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">ESeq</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><span class="kw"> such that </span><br/><a NAME="E16:47_1_1_3"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/> ( <font color="Maroon" title="c13">DSeq</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="c14">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="c14">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#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#T33" target="_self" onmouseover="rs('prob_4/T33')" onmouseout="rh()">Th33</a></span>;<br/>
<a NAME="E17:47_1_1_3"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">BSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E18:47_1_1_3"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon" title="c12">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">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/>

<span class="kw">assume </span><a NAME="E1:47_1_1_3_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:47_1_1_3_3"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:47_1_1_3_3"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<font color="Maroon" title="c16">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c10">CSeq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_1_1_3"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:47_1_1_3_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">CSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:47_1_1_3_3"/><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E8:47_1_1_3_3"/><span class="lab"><font color="Green" title="E50">A19</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_3_3"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">n</font> = <font color="Maroon" title="c17">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_1_3_3"><span class="lab"><font color="Green" title="E49">A18</font></span></a></span>;<br/>
<span class="kw">set </span><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="E10:47_1_1_3_3"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1_1_3"><span class="lab"><font color="Green" title="E43">A12</font></span></a></span>;<br/>
<a NAME="E11:47_1_1_3_3"/>
 ex <font color="Olive" title="b1">P</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">P</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R2" title="TARSKI: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/>

<span class="kw">thus </span><a NAME="E1:47_1_1_3_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="tarski.html#R2" title="TARSKI: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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47_1_1_3"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E3:47_1_1_3_3"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E8:47_1_1_3_3"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E10:47_1_1_3_3"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:47_1_1_3_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:47_1_1_3"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
 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="c12">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><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="c12">B</font>;<br/>
<span class="lab"><font color="Green" title="E48">A22</font></span>:  <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="c12">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="c12">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T51" onmouseover="rs('xboole_1/T51')" onmouseout="rh()">XBOOLE_1:51</a></span>
<br/>.= 
<font color="Maroon" title="c12">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="c12">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:47_1_1_3"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">reconsider </span><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="c12">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="E3:47_1_1_3_4"/><span class="lab"><font color="Green" title="E49">A23</font></span>: 
<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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c13">DSeq</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c13">DSeq</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_3_4_2"/><span class="lab"><font color="Green" title="E50">A24</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="c13">DSeq</font></span>)</span></span><br/>

<a NAME="E2:47_1_1_3_4_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:47_1_1_3_4_2"/><span class="lab"><font color="Green" title="E51">A25</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:47_1_1_3_4_2"/><span class="lab"><font color="Green" title="E52">A26</font></span>: 
<font color="Maroon" title="c18">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c9">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:47_1_1_3_4_2"/><span class="lab"><font color="Green" title="E53">A27</font></span>: 
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E8:47_1_1_3_4_2"/><span class="lab"><font color="Green" title="E54">A28</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47_1_1_3"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E5:47_1_1_3_4_2"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_1_3_4_2"><span class="lab"><font color="Green" title="E53">A27</font></span></a></span>;<br/>
<a NAME="E10:47_1_1_3_4_2"/><span class="lab"><font color="Green" title="E55">A29</font></span>: 
 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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c10">CSeq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_1_1_3"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E1:47_1_1_3_4_2"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E11:47_1_1_3_4_2"/>
 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_4_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:47_1_1_3_4_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">CSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:47_1_1_3_4_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A30</font></span>: 
<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c10">CSeq</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:47_1_1_3_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="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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E4:47_1_1_3_4_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_1_3_4_2"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="txt" href="#E2:47_1_1_3_4_2_1"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:47_1_1_3_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A31</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_3_4_2"><span class="lab"><font color="Green" title="E51">A25</font></span></a>, <a class="txt" href="#E8:47_1_1_3_4_2"><span class="lab"><font color="Green" title="E54">A28</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E13:47_1_1_3_4_2"/>
 ex <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c13">DSeq</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_4_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c13">DSeq</font> )</span><br/>

<a NAME="E1:47_1_1_3_4_2_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">DSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:47_1_1_3_4_2_2"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c13">DSeq</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47_1_1_3"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E12:47_1_1_3_4_2"><span class="lab"><font color="Green" title="E56">A31</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:47_1_1_3_4_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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c13">DSeq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:47_1_1_3_4"/>
 for <font color="Olive" title="b1">A</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c14">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#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">A</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c14">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#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<a NAME="E1:47_1_1_3_4_3"/><span class="lab"><font color="Green" title="E50">A32</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">ESeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:47_1_1_3_4_3"/>
<font color="Maroon" title="c17">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c14">ESeq</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">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#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E3:47_1_1_3_4_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c17">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ESeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_3_4_3"><span class="lab"><font color="Green" title="E50">A32</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:47_1_1_3_4_3"/>
<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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:47_1_1_3"><span class="lab"><font color="Green" title="E44">A13</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:47_1_1_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A33</font></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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c14">ESeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#T8" target="_self" onmouseover="rs('prob_4/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<a NAME="E6:47_1_1_3_4"/>
 <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c14">ESeq</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T17" onmouseover="rs('prob_1/T17')" onmouseout="rh()">PROB_1:17</a></span>;<br/>
<a NAME="E7:47_1_1_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A34</font></span>: 
 <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="c14">ESeq</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<a NAME="E8:47_1_1_3_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="c13">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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c14">ESeq</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_4_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="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="c13">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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c14">ESeq</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_3_4_4"/>
<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="c13">DSeq</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="c14">ESeq</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:47_1_1_3_4_4"/><span class="lab"><font color="Green" title="E52">A35</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:47_1_1_3_4_4"/><span class="lab"><font color="Green" title="E53">A36</font></span>: 
<font color="Maroon" title="c18">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c13">DSeq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:47_1_1_3_4_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">DSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:47_1_1_3_4_4"/><span class="lab"><font color="Green" title="E54">A37</font></span>: 
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E8:47_1_1_3_4_4"/><span class="lab"><font color="Green" title="E55">A38</font></span>: 
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">DSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_3_4_4"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_1_3_4_4"><span class="lab"><font color="Green" title="E54">A37</font></span></a></span>;<br/>
<a NAME="E10:47_1_1_3_4_4"/>
<font color="Maroon" title="c20">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">ESeq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_1_3_4_4"><span class="lab"><font color="Green" title="E54">A37</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:47_1_1_3_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A39</font></span>: 
<font color="Maroon" title="c14">ESeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c14">ESeq</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E12:47_1_1_3_4_4"/>
<font color="Maroon" title="c13">DSeq</font> <a href="prob_4.html#K1" title="PROB_4: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="c14">ESeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <font color="Maroon" title="c20">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:47_1_1_3"><span class="lab"><font color="Green" title="E44">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:47_1_1_3_4_4"/>
<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="c14">ESeq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_3_4_4"><span class="lab"><font color="Green" title="E52">A35</font></span></a>, <a class="txt" href="#E8:47_1_1_3_4_4"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E11:47_1_1_3_4_4"><span class="lab"><font color="Green" title="E56">A39</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:47_1_1_3_4"/><span class="kw">then </span>
<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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c14">ESeq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_3_4"><span class="lab"><font color="Green" title="E49">A23</font></span></a></span>;<br/>
<a NAME="E10:47_1_1_3_4"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_1_3_4"><span class="lab"><font color="Green" title="E51">A34</font></span></a>, <a class="txt" href="#E5:47_1_1_3_4"><span class="lab"><font color="Green" title="E50">A33</font></span></a>, <a class="ref" href="prob_4.html#D4" target="_self" onmouseover="rs('prob_4/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E12:47_1_1_3_4"/><span class="lab"><font color="Green" title="E52">A40</font></span>: 
 <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="c12">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c17">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_3_4"><span class="lab"><font color="Green" title="E48">A22</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">C</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c12">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c17">C</font></span><br/>

<span class="kw">thus </span><a NAME="E13:47_1_1_3_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="c12">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c17">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47_1_1_3_4"><span class="lab"><font color="Green" title="E52">A40</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:47_1_1_3"/>
 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">object</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">n</font>, <font color="Maroon" title="c16">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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/>

<span class="lab"><font color="Green" title="E48">A41</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">F</font> = 
 <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">CSeq</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E2:47_1_1_3_5"/>
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="47_1_1_3_5_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1_1_3_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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_3_5_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_3_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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">F</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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"><span class="kw">hence </span><a NAME="E2:47_1_1_3_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1_1_3"><span class="lab"><font color="Green" title="E43">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_3_5_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_3_5_2_1_2"/><span class="lab"><font color="Green" title="E49">A42</font></span>: 
 not <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">F</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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"><span class="kw">then </span><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> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_3_5"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="txt" href="#E1:47_1_1_3_5_2_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:47_1_1_3_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:47_1_1_3_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A43</font></span>: 
( <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/>
<span class="kw">assume </span><a NAME="E4:47_1_1_3_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="E5:47_1_1_3_5"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_3"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="prob_2.html#D2" onmouseover="rs('prob_2/D2')" onmouseout="rh()">PROB_2:def 2</a></span>;<br/>
<a NAME="E6:47_1_1_3_5"/><span class="kw">then </span>
<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> 
 
;<br/>
<a NAME="E7:47_1_1_3_5"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_3_5"><span class="lab"><font color="Green" title="E49">A43</font></span></a>, <a class="ref" href="xboole_1.html#T3" onmouseover="rs('xboole_1/T3')" onmouseout="rh()">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:47_1_1_3_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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E21:47_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A44</font></span>: 
<font color="Maroon" title="c10">CSeq</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">V128</a>()
 
;<br/>
<a NAME="E22:47_1_1_3"/>
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_6"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:47_1_1_3_6"/><span class="lab"><font color="Green" title="E49">A45</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c6">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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="E1:47_1_1_3_6_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1_1_3"><span class="lab"><font color="Green" title="E43">A12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:47_1_1_3_6_1"/><span class="lab"><font color="Green" title="E50">A46</font></span>: 
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T51" onmouseover="rs('xboole_1/T51')" onmouseout="rh()">XBOOLE_1:51</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1_1_3"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E3:47_1_1_3_6_1"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:47_1_1_3_6_1"/>
<font color="Maroon" title="c6">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:47_1_1_3_6"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="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#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">CSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="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#K8" title="PROB_1:func.8">*</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="c15">x</font></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c15">x</font> = 
<font color="Maroon" title="c6">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">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<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="c15">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_3_6"><span class="lab"><font color="Green" title="E49">A45</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</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="c15">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:47_1_1_3_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="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#K8" title="PROB_1:func.8">*</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="c15">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:47_1_1_3_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A47</font></span>: 
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">CSeq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>;<br/>
<a NAME="E4:47_1_1_3_6"/>
<font color="Maroon" title="c6">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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c10">CSeq</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E8:47_1_1_3"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E19:47_1_1_3"><span class="lab"><font color="Green" title="E47">A21</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c6">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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c10">CSeq</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>
<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#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c10">CSeq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:47_1_1_3_6"/>
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:47_1_1_3"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E3:47_1_1_3_6"><span class="lab"><font color="Green" title="E50">A47</font></span></a>, <a class="ref" href="prob_3.html#T46" onmouseover="rs('prob_3/T46')" onmouseout="rh()">PROB_3:46</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E23:47_1_1_3"/>
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:47_1_1"/><span class="lab"><font color="Green" title="E39">A48</font></span>: 
 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#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) holds  <font color="Maroon" title="c6">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#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:47_1_1_4"/><span class="lab"><font color="Green" title="E40">A49</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">and </span><br/><a NAME="E3:47_1_1_4"/><span class="lab"><font color="Green" title="E41">A50</font></span>: 
 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="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#D5" target="_self" onmouseover="rs('prob_4/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c8">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1_4"><span class="lab"><font color="Green" title="E40">A49</font></span></a></span>;<br/>
<a NAME="E5:47_1_1_4"/>
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">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="c9">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E3:47_1_1_4"><span class="lab"><font color="Green" title="E41">A50</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:47_1_1_4"/>
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D8" onmouseover="rs('prob_1/D8')" onmouseout="rh()">PROB_1:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:47_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="c4">C</font>
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E40">A51</font></span>: <font color="Maroon" title="c6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="prob_1.html#T4" onmouseover="rs('prob_1/T4')" onmouseout="rh()">PROB_1:4</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" onmouseover="rs('valued_0/D19')" onmouseout="rh()">VALUED_0:def 19</a></span>
;<br/>
<a NAME="E11:47_1_1"/><span class="lab"><font color="Green" title="E41">A52</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">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="c6">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">A</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">B</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="c6">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">A</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_6"/><span class="lab"><font color="Green" title="E42">A53</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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">A</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">A1</font> = <font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c10">B1</font> = <font color="Maroon" title="c8">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">BSeq</font> = (<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">B1</font>) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">Omega</font></span>)</span> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font> ;<br/>
<a NAME="E4:47_1_1_6"/><span class="lab"><font color="Green" title="E43">A54</font></span>: 
<font color="Maroon" title="c11">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T123" onmouseover="rs('funct_7/T123')" onmouseout="rh()">FUNCT_7:123</a></span>;<br/>
<a NAME="E5:47_1_1_6"/><span class="lab"><font color="Green" title="E44">A55</font></span>: 
<font color="Maroon" title="c11">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T122" onmouseover="rs('funct_7/T122')" onmouseout="rh()">FUNCT_7:122</a></span>;<br/>
<a NAME="E6:47_1_1_6"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">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="47_1_1_6_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1_1_6_1_1"/>
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_6_1_1_1"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:47_1_1_6_1_1_1"/>
<font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_1_6"><span class="lab"><font color="Green" title="E44">A55</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_6_1_1_2"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)</span><br/><div class="add"><a NAME="E2:47_1_1_6_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E3:47_1_1_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A56</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_6_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1_1_6_1_1_2_1"/>
( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_6_1_1_2"><span class="lab"><font color="Green" title="E45">A56</font></span></a>, <a class="ref" href="nat_1.html#T22" onmouseover="rs('nat_1/T22')" onmouseout="rh()">NAT_1:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_6_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_6_1_1_2_1_1"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)</span><br/><div class="add"><a NAME="E2:47_1_1_6_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T124" onmouseover="rs('funct_7/T124')" onmouseout="rh()">FUNCT_7:124</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_1_1_6_1_1_2_1_1"/>
<font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T4" onmouseover="rs('prob_1/T4')" onmouseout="rh()">PROB_1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_6_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_6_1_1_2_1_2"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:47_1_1_6_1_1_2_1_2"/>
<font color="Maroon" title="c11">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_6"><span class="lab"><font color="Green" title="E43">A54</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E7:47_1_1_6"/><span class="kw">then </span>
 <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c11">BSeq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T52" onmouseover="rs('nat_1/T52')" onmouseout="rh()">NAT_1:52</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">BSeq</font> = <font color="Maroon" title="c11">BSeq</font> as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E9:47_1_1_6"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_6_2"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E45">A57</font></span>: <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1_1_6"><span class="lab"><font color="Green" title="E44">A55</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<a NAME="E2:47_1_1_6_2"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E46">A58</font></span>: <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">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#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_6"><span class="lab"><font color="Green" title="E43">A54</font></span></a>, <a class="txt" href="#E1:47_1_1_6_2"><span class="lab"><font color="Green" title="E45">A57</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<a NAME="E4:47_1_1_6_2"/><span class="lab"><font color="Green" title="E47">A59</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_6_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span>;<br/>
<a NAME="E1:47_1_1_6_2_3"/><span class="lab"><font color="Green" title="E48">A60</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_6_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_6_2_3_1"/><span class="lab"><font color="Green" title="E49">A61</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:47_1_1_6_2_3_1"/><span class="lab"><font color="Green" title="E50">A62</font></span>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">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#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_6_2_3_1"><span class="lab"><font color="Green" title="E49">A61</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1_6_2_3_1"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="ref" href="funct_7.html#T124" onmouseover="rs('funct_7/T124')" onmouseout="rh()">FUNCT_7:124</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_1"><span class="lab"><font color="Green" title="E40">A51</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:47_1_1_6_2_3"/>
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">BSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> 2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_6_2"><span class="lab"><font color="Green" title="E46">A58</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T124" onmouseover="rs('funct_7/T124')" onmouseout="rh()">FUNCT_7:124</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_1"><span class="lab"><font color="Green" title="E40">A51</font></span></a></span>
;<br/>
<a NAME="E4:47_1_1_6_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A63</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<span class="kw">thus </span><a NAME="E5:47_1_1_6_2_3"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:47_1_1_6_2_3"><span class="lab"><font color="Green" title="E49">A63</font></span></a>, <a class="txt" href="#E1:47_1_1_6_2_3"><span class="lab"><font color="Green" title="E48">A60</font></span></a>);</span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c13">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> implies <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E5:47_1_1_6_2"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:47_1_1_6_2"/><span class="lab"><font color="Green" title="E48">A64</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:47_1_1_6_2"/>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_6_2"><span class="lab"><font color="Green" title="E47">A59</font></span></a>, <a class="txt" href="#E7:47_1_1_6_2"><span class="lab"><font color="Green" title="E48">A64</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:47_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A65</font></span>: 
(  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c12">BSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c8">B</font> &amp;  <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c12">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#T3" onmouseover="rs('dynkin/T3')" onmouseout="rh()">DYNKIN:3</a>, <a class="ref" href="prob_1.html#T1" onmouseover="rs('prob_1/T1')" onmouseout="rh()">PROB_1:1</a></span>;<br/>
<a NAME="E11:47_1_1_6"/>
<font color="Maroon" title="c12">BSeq</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">V128</a>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_6"><span class="lab"><font color="Green" title="E42">A53</font></span></a>, <a class="ref" href="prob_4.html#T3" target="_self" onmouseover="rs('prob_4/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:47_1_1_6"/>
<font color="Maroon" title="c6">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">A</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E10:47_1_1_6"><span class="lab"><font color="Green" title="E45">A65</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:47_1_1"/><span class="lab"><font color="Green" title="E42">A66</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_7"/><span class="lab"><font color="Green" title="E43">A67</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c6">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">A</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c7">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47_1_1"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>
<br/>.= 
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_7"><span class="lab"><font color="Green" title="E43">A67</font></span></a>, <a class="ref" href="xboole_1.html#T45" onmouseover="rs('xboole_1/T45')" onmouseout="rh()">XBOOLE_1:45</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:47_1_1_7"/>
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:47_1_1"/><span class="lab"><font color="Green" title="E43">A68</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">B</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_8"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font></span><br/>

<a NAME="E2:47_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47_1_1"><span class="lab"><font color="Green" title="E42">A66</font></span></a></span>;<br/>
<a NAME="E3:47_1_1_8"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_1_1"><span class="lab"><font color="Green" title="E39">A48</font></span></a></span>;<br/>
<a NAME="E4:47_1_1_8"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:47_1_1_8"/>
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:47_1_1"/><span class="lab"><font color="Green" title="E44">A69</font></span>: 
 for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">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-descending</a>  holds <br/><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">BSeq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">BSeq</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">BSeq</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  implies <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_9"/><span class="lab"><font color="Green" title="E45">A70</font></span>: 
<font color="Maroon" title="c7">BSeq</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> </span><br/>

<a NAME="E2:47_1_1_9"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">n</font>, <font color="Maroon" title="c9">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" 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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font> )</span><br/>

<a NAME="E1:47_1_1_9_1"/><span class="lab"><font color="Green" title="E46">A71</font></span>: 
( <font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c9">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:47_1_1_9_1"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span><br/>

<a NAME="E3:47_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">BSeq</font> <a href="prob_2.html#K1" title="PROB_2: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_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c9">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_9"><span class="lab"><font color="Green" title="E45">A70</font></span></a>, <a class="ref" href="prob_1.html#D5" onmouseover="rs('prob_1/D5')" onmouseout="rh()">PROB_1:def 5</a></span>;<br/>
<a NAME="E4:47_1_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">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_2.html#K1" title="PROB_2: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="c6">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_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1_1"><span class="lab"><font color="Green" title="E43">A68</font></span></a></span>;<br/>
<a NAME="E5:47_1_1_9_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c6">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_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a>, <a class="txt" href="#E1:47_1_1_9_1"><span class="lab"><font color="Green" title="E46">A71</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:47_1_1_9_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a>, <a class="txt" href="#E1:47_1_1_9_1"><span class="lab"><font color="Green" title="E46">A71</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:47_1_1_9"/>
<font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#T6" onmouseover="rs('seqm_3/T6')" onmouseout="rh()">SEQM_3:6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E45">A72</font></span>: <font color="Maroon" title="c6">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="c6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="prob_1.html#T5" onmouseover="rs('prob_1/T5')" onmouseout="rh()">PROB_1:5</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D8" onmouseover="rs('prob_1/D8')" onmouseout="rh()">PROB_1:def 8</a></span>
;<br/>
<a NAME="E16:47_1_1"/><span class="lab"><font color="Green" title="E46">A73</font></span>: 
 for <font color="Olive" title="b1">A</font> being    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) holds  <font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c1">Omega</font> as    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T23" onmouseover="rs('prob_1/T23')" onmouseout="rh()">PROB_1:23</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">A</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>

<a NAME="E2:47_1_1_11"/>
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47_1_1"><span class="lab"><font color="Green" title="E43">A68</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:47_1_1_11"/>
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:47_1_1"><span class="lab"><font color="Green" title="E45">A72</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:47_1_1"/><span class="lab"><font color="Green" title="E47">A74</font></span>: 
 for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">BSeq</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</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/><span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</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="E1:47_1_1_12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:47_1_1_12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E16:47_1_1"><span class="lab"><font color="Green" title="E46">A73</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:47_1_1"/><span class="lab"><font color="Green" title="E48">A75</font></span>: 
 for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">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-descending</a>  holds <br/>( <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">BSeq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">V111</a>() &amp; <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">BSeq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">BSeq</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">BSeq</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  implies ( <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">V111</a>() &amp; <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_13"/>
<font color="Maroon" title="c7">BSeq</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">V111</a>() &amp; <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )</span><br/>

<a NAME="E2:47_1_1_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A76</font></span>: 
<font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:47_1_1"><span class="lab"><font color="Green" title="E44">A69</font></span></a></span>;<br/>
<a NAME="E3:47_1_1_13"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="E1:47_1_1_13_1"/>
<font color="Maroon" title="c8">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:47_1_1_13_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47_1_1"><span class="lab"><font color="Green" title="E47">A74</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:47_1_1_13_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:47_1_1_13"/>
<font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">V111</a>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D3" onmouseover="rs('seq_2/D3')" onmouseout="rh()">SEQ_2:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> </span><br/>

<span class="kw">hence </span><a NAME="E5:47_1_1_13"/>
<font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1_13"><span class="lab"><font color="Green" title="E49">A76</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:47_1_1"/>
 for <font color="Olive" title="b1">BSeq</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">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-descending</a>  holds <br/>( <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">BSeq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b1">BSeq</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">BSeq</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">BSeq</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  implies ( <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_14"/><span class="lab"><font color="Green" title="E49">A77</font></span>: 
<font color="Maroon" title="c7">BSeq</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">CSeq</font> =  <a href="prob_3.html#K3" title="PROB_3:func.3">Partial_Diff_Union</a> <font color="Maroon" title="c7">BSeq</font> as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) ;<br/>
<a NAME="E3:47_1_1_14"/><span class="lab"><font color="Green" title="E50">A78</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E1:47_1_1_14_1"/><span class="lab"><font color="Green" title="E51">A79</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_14_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_14_1_1"/><span class="lab"><font color="Green" title="E52">A80</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:47_1_1_14_1_1"/><span class="lab"><font color="Green" title="E53">A81</font></span>: 
<font color="Maroon" title="c7">BSeq</font> <a href="prob_2.html#K1" title="PROB_2: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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_14"><span class="lab"><font color="Green" title="E49">A77</font></span></a>, <a class="ref" href="prob_2.html#T7" onmouseover="rs('prob_2/T7')" onmouseout="rh()">PROB_2:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">kk</font> = <font color="Maroon" title="c9">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</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="c10">kk</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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_14_1_1"><span class="lab"><font color="Green" title="E52">A80</font></span></a>, <a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c10">kk</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="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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_2.html#K1" title="PROB_2: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="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">CSeq</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c6">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_2.html#K1" title="PROB_2: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="c8">CSeq</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47_1_1"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="txt" href="#E1:47_1_1_14"><span class="lab"><font color="Green" title="E49">A77</font></span></a>, <a class="ref" href="prob_4.html#T18" target="_self" onmouseover="rs('prob_4/T18')" onmouseout="rh()">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c6">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_2.html#K1" title="PROB_2: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#K1" title="NAT_1:func.1">+</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_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_14"><span class="lab"><font color="Green" title="E49">A77</font></span></a>, <a class="ref" href="prob_4.html#T16" target="_self" onmouseover="rs('prob_4/T16')" onmouseout="rh()">Th16</a></span>
<br/>.= 
<font color="Maroon" title="c6">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_2.html#K1" title="PROB_2: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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" onmouseover="rs('xboole_1/T39')" onmouseout="rh()">XBOOLE_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c6">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1_14_1_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c6">comP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">CSeq</font> <a href="prob_4.html#K1" title="PROB_4:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c6">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#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_14"><span class="lab"><font color="Green" title="E49">A77</font></span></a>, <a class="ref" href="prob_4.html#T16" target="_self" onmouseover="rs('prob_4/T16')" onmouseout="rh()">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</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#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<a NAME="E3:47_1_1_14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A82</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<span class="kw">thus </span><a NAME="E4:47_1_1_14_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E3:47_1_1_14_1"><span class="lab"><font color="Green" title="E52">A82</font></span></a>, <a class="txt" href="#E1:47_1_1_14_1"><span class="lab"><font color="Green" title="E51">A79</font></span></a>);</span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:47_1_1_14"/><span class="lab"><font color="Green" title="E51">A83</font></span>: 
 <a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_14"><span class="lab"><font color="Green" title="E50">A78</font></span></a></span>;<br/>
<font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> = 
<font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c8">CSeq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T36" onmouseover="rs('prob_3/T36')" onmouseout="rh()">PROB_3:36</a></span>
<br/>.= 
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a></span>
<br/>.= 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c8">CSeq</font></span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E6:47_1_1_14"/>
( <font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">comP</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:47_1_1"><span class="lab"><font color="Green" title="E48">A75</font></span></a>, <a class="txt" href="#E1:47_1_1_14"><span class="lab"><font color="Green" title="E49">A77</font></span></a>, <a class="txt" href="#E4:47_1_1_14"><span class="lab"><font color="Green" title="E51">A83</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">comP</font> = <font color="Maroon" title="c6">comP</font> as    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_1_1"><span class="lab"><font color="Green" title="E39">A48</font></span></a>, <a class="txt" href="#E15:47_1_1"><span class="lab"><font color="Green" title="E45">A72</font></span></a>, <a class="txt" href="#E11:47_1_1"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="ref" href="prob_2.html#T10" onmouseover="rs('prob_2/T10')" onmouseout="rh()">PROB_2:10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">comP</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI: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/>

<span class="kw">thus </span><a NAME="E21:47_1_1"/>
 for <font color="Olive" title="b1">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI: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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
