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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>  st <font color="Olive" title="b1">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  holds <br/> for <font color="Olive" title="b2">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font> holds   <a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b1">f</font>) is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  implies  for <font color="Olive" title="b1">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font> holds   <a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">f</font>) is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:16"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font> holds   <a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">f</font>) is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">X</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>) is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> </span><br/>

<a NAME="E2:16"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> implies <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:16_1"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 ; <a class="txt" 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="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font></span><br/>

<a NAME="E2:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="prob_2.html#D2" title="PROB_2:def.2">PROB_2:def 2</a></span>;<br/>
<a NAME="E3:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T74" title="XBOOLE_1:th.74">XBOOLE_1:74</a></span>;<br/>
<a NAME="E4:16_1"/>
( <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" target="_self" title="DYNKIN:def.1">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_1"/>
<span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="xboole_1.html#T74" title="XBOOLE_1:th.74">XBOOLE_1:74</a></span>; <a class="txt" 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="E3:16"/>
 <a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">f</font>) is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D2" target="_self" title="DYNKIN:def.2">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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