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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">ASeq</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">Sigma</font><br/> for <font color="Olive" title="b3">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="b3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b2">ASeq</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b2">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b2">ASeq</font></span>)</span></span>)</span> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b2">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Olive" title="b2">ASeq</font></span>)</span> )</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">ASeq</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</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="Maroon" title="c2">Sigma</font> holds <br/> ( <font color="Olive" title="b2">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b1">ASeq</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b1">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b1">ASeq</font></span>)</span></span>)</span> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Olive" title="b1">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Olive" title="b1">ASeq</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">ASeq</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</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"> 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_1.html#K9" title="PROB_1:func.9">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> )</span><br/>







<a NAME="E1:37"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> 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="prob_3.html#T42" target="_self" title="PROB_3:th.42">Th42</a></span>;<br/>
<a NAME="E2:37"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T13" target="_self" title="PROB_3:th.13">Th13</a></span>;<br/>
<a NAME="E3:37"/><span class="kw">then </span>
<font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_2.html#T20" title="PROB_2:th.20">PROB_2:20</a></span>;<br/>
<a NAME="E4:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T27" title="SEQ_2:th.27">SEQ_2:27</a></span>;<br/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="prob_2.html#T20" title="PROB_2:th.20">PROB_2:20</a></span>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T17" target="_self" title="PROB_3:th.17">Th17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:37"/>
( <font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" title="PROB_3:func.5">@Partial_Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">ASeq</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="prob_2.html#T20" title="PROB_2:th.20">PROB_2:20</a>, <a class="ref" href="rinfsup1.html#T24" title="RINFSUP1:th.24">RINFSUP1:24</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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