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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

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for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="prob_1.html#V3">non-decreasing</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> ) holds <br/> <a href="kurato_2.html#K1">Union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E2:72"/>
 <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> is <a href="prob_3.html#V2" target="_self">non-decreasing-closed</a>
 <b>by </b><i><a class="ref" href="prob_3.html#D12" target="_self">Def12</a></i>;<br/>

<a NAME="E3:72"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="prob_1.html#V2">non-increasing</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> ) holds <br/> <a href="prob_1.html#K4">Intersection</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E4:72"/>
 <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> is <a href="prob_3.html#V3" target="_self">non-increasing-closed</a>
 <b>by </b><i><a class="ref" href="prob_3.html#D13" target="_self">Def13</a></i>;<br/>


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