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<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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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 <br/> ( <font color="Olive" title="b2">P</font> <a href="prob_4.html#R1" title="PROB_4:pred.1">is_complete</a> <font color="Olive" title="b1">Sigma</font> iff  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Olive" title="b2">P</font> <a href="measure3.html#R1" title="MEASURE3:pred.1">is_complete</a> <font color="Olive" title="b1">Sigma</font> )</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <br/> ( <font color="Olive" title="b1">P</font> <a href="prob_4.html#R1" title="PROB_4:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font> iff  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Olive" title="b1">P</font> <a href="measure3.html#R1" title="MEASURE3:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font> )</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">P</font> <a href="prob_4.html#R1" title="PROB_4:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font> iff  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> <a href="measure3.html#R1" title="MEASURE3:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font> )</span><br/>





<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> <a href="measure3.html#R1" title="MEASURE3:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font> implies <font color="Maroon" title="c3">P</font> <a href="prob_4.html#R1" title="PROB_4:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:29_1"/>
<font color="Maroon" title="c3">P</font> <a href="prob_4.html#R1" title="PROB_4:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> <a href="measure3.html#R1" title="MEASURE3:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font></span><br/><a NAME="E2:29_1"/><span class="kw">then </span>
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#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</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#K2" title="PROB_4:func.2">P2M</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="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#D3" target="_self" title="PROB_4:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:29_1"/>
 <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> <a href="measure3.html#R1" title="MEASURE3:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#D2" title="MEASURE3:def.2">MEASURE3:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:29"/>
 <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> <a href="measure3.html#R1" title="MEASURE3:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">P</font> <a href="prob_4.html#R1" title="PROB_4:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font></span><br/>

<a NAME="E3:29"/><span class="kw">then </span>
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#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</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; <font color="Maroon" title="c3">P</font> <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#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#D2" title="MEASURE3:def.2">MEASURE3:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:29"/>
<font color="Maroon" title="c3">P</font> <a href="prob_4.html#R1" title="PROB_4:pred.1">is_complete</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#D3" target="_self" title="PROB_4:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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