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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="Red">::  thesis: </font></span></a><span class="hide"> 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">X</font><br/> for <font color="Olive" title="b2">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>  holds <br/><font color="Olive" title="b1">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Olive" title="b2">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</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">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>  holds <br/><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Olive" title="b1">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>  implies <font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:31"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c3">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> 
 ; <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="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_1"/>
( <font color="Maroon" title="c3">A1</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  or <font color="Maroon" title="c3">A1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="setlim_1.html#D1" title="SETLIM_1:def.1">SETLIM_1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><span class="kw">suppose </span></a><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c3">A1</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> 
 ; <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="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> </span><br/><div class="add"><a NAME="E2:31_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</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="setlim_2.html#T27" target="_self" title="SETLIM_2:th.27">Th27</a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#D1" title="SETLIM_1:def.1">SETLIM_1:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><span class="kw">suppose </span></a><a NAME="E1:31_1_2"/>
<font color="Maroon" title="c3">A1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 ; <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="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> </span><br/><div class="add"><a NAME="E2:31_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">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="setlim_2.html#T28" target="_self" title="SETLIM_2:th.28">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#D1" title="SETLIM_1:def.1">SETLIM_1:def 1</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></div><span class="kw">end;</span></div>

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