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

<span class="kw">let </span><font color="Maroon" title="c1">Omega</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Sigma</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font><br/>  for <font color="Olive" title="b2">P</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Olive" title="b1">Sigma</font> holds   <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Olive" title="b2">P</font> is  <a href="prob_4.html#V1" title="PROB_4:attr.1">complete</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</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">P</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font> holds   <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Olive" title="b1">P</font> is  <a href="prob_4.html#V1" title="PROB_4:attr.1">complete</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <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">  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c3">P</font> is  <a href="prob_4.html#V1" title="PROB_4:attr.1">complete</a> </span><br/>





<a NAME="E1:49"/>
 for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font><br/>  for <font color="Olive" title="b2">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">B</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">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> &amp; <span class="p1">(<span class="default"><a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Olive" title="b1">A</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="49_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</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>  <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="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp; <span class="p1">(<span class="default"><a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Maroon" title="c4">A</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/><span class="kw">let </span><font color="Maroon" title="c5">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="c5">B</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="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> &amp; <span class="p1">(<span class="default"><a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c4">A</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/>



<span class="kw">assume </span><a NAME="E1:49_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c5">B</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>)
 ; <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="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> or  not <span class="p1">(<span class="default"><a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</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="c4">A</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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:49_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E3:49_1"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<span class="p1">(<span class="default"><a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B</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="c4">A</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/>

<a NAME="E4:49_1"/>
 ex <font color="Olive" title="b1">B1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">B1</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="b2">C1</font> being    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> st <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C1</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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">C1</font> being    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> st <font color="Maroon" title="c4">A</font> <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="Olive" title="b1">C1</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">B2</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:49_1_1"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon" title="c6">B2</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="E3:49_1_1"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 ex <font color="Olive" title="b1">C2</font> being    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> st <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="prob_4.html#D5" target="_self" onmouseover="rs('prob_4/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E4:49_1_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">B2</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="#E3:49_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E2:49_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E3:49_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="prob_4.html#D8" target="_self" onmouseover="rs('prob_4/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">C2</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="E6:49_1_1"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">C1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">D2</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E8:49_1_1"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<font color="Maroon" title="c9">D2</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="E9:49_1_1"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c7">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">D2</font>
 <span class="kw">and </span><br/><a NAME="E10:49_1_1"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#D4" target="_self" onmouseover="rs('prob_4/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">O</font> = <font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">D2</font>;<br/>
<a NAME="E11:49_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E12:49_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A11</font></span>: 
( <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B2</font> &amp; <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">D2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:49_1_1"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E13:49_1_1"/>
 ex <font color="Olive" title="b1">O</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">O</font> &amp; <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O</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="49_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c11">B2</font> = <font color="Maroon" title="c6">B2</font>, <font color="Maroon" title="c12">D2</font> = <font color="Maroon" title="c9">D2</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="#E2:49_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E8:49_1_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">D2</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">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">D2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">D2</font> &amp; <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"><font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">D2</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><br/>

<a NAME="E2:49_1_1_1"/>
<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="c11">B2</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c12">D2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.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="#E4:49_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E10:49_1_1"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="prob_1.html#T39" onmouseover="rs('prob_1/T39')" onmouseout="rh()">PROB_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:49_1_1_1"/>
( <font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">D2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">D2</font> &amp; <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"><font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">D2</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="txt" href="#E12:49_1_1"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="prob_1.html#D8" onmouseover="rs('prob_1/D8')" onmouseout="rh()">PROB_1:def 8</a>, <a class="ref" href="xboole_1.html#T13" onmouseover="rs('xboole_1/T13')" onmouseout="rh()">XBOOLE_1:13</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="E14:49_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</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#D4" target="_self" onmouseover="rs('prob_4/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<font color="Maroon" title="c4">A</font> = 
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E6:49_1_1"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <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="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" onmouseover="rs('xboole_1/T23')" onmouseout="rh()">XBOOLE_1:23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:49_1_1"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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">C1</font> being    <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> st <font color="Maroon" title="c4">A</font> <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="Olive" title="b1">C1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:49_1_1"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <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>
<span class="kw">hence </span><a NAME="E5:49_1"/>
<font color="Maroon" title="c4">A</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_4.html#D5" target="_self" onmouseover="rs('prob_4/D5')" onmouseout="rh()">Def5</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="E2:49"/>
 <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c3">P</font> is  <a href="prob_4.html#V1" title="PROB_4:attr.1">complete</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"> verum</span><br/>


</div>
