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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E35">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></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-increasing</a> or <font color="Maroon" title="c3">A1</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> )
 </span><b>by </b><i><a class="txt" href="setlim_2.html#E1:31"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="setlim_1.html#D1" title="SETLIM_1:def.1">SETLIM_1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>suppose </b></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-increasing</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
<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-decreasing</a>
 <b>by </b><i><a class="ref" href="setlim_2.html#T27" target="_self" title="SETLIM_2:th.27">Th27</a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="ref" href="setlim_1.html#D1" title="SETLIM_1:def.1">SETLIM_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><b>suppose </b></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-decreasing</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
<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-increasing</a>
 <b>by </b><i><a class="ref" href="setlim_2.html#T28" target="_self" title="SETLIM_2:th.28">Th28</a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="ref" href="setlim_1.html#D1" title="SETLIM_1:def.1">SETLIM_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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