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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <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">A1</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> holds   <a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Olive" title="b1">A1</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c2">A1</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> </span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> <span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><a NAME="E1:13_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#D1" target="_self" title="PROB_3:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E2:13_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</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="E2:13"/>
 <a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c2">A1</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_2.html#T6" title="PROB_2:th.6">PROB_2:6</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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