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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">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">SSets</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b3">SSets</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  holds <br/><font color="Olive" title="b2">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Olive" title="b3">SSets</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</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">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">SSets</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">SSets</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  holds <br/><font color="Olive" title="b1">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Olive" title="b2">SSets</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</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">SSets</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">SSets</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  holds <br/><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Olive" title="b1">SSets</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">SSets</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">S</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="c4">SSets</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  implies <font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  )</span><br/>







<a NAME="E1:53"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:53"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c4">SSets</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="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">n</font>, <font color="Maroon" title="c6">m</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font> )</span><br/><a NAME="E1:53_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
( <font color="Maroon" title="c4">SSets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">SSets</font> &amp; <font color="Maroon" title="c4">SSets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">SSets</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/><a NAME="E2:53_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">SSets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/><span class="kw">assume </span><a NAME="E3:53_1"/>
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font></span><br/><a NAME="E4:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c4">SSets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">SSets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="prob_1.html#D6" title="PROB_1:def.6">PROB_1:def 6</a></span>;<br/><a NAME="E5:53_1"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">SSets</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">SSets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E6:53_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</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>
<span class="kw">hence </span><a NAME="E4:53"/>
<font color="Maroon" title="c3">M</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c4">SSets</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T7" title="RINFSUP2:th.7">RINFSUP2:7</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/>


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