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<b>let </b><font color="Maroon" title="c1">Omega</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">ASeq</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font> holds <br/> ( <font color="Olive" title="b1">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> iff  <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Olive" title="b1">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> )</span><br/><b>let </b><font color="Maroon" title="c2">ASeq</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> iff  <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> )</span><br/>



<b>thus </b><a NAME="E1:12"/>
( <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> implies  <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> implies <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</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="12_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> implies <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> )</span><br/><b>assume </b><a NAME="E1:12_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span><br/><a NAME="E2:12_1_1"/><b>then </b>
<font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="#E1:12_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prob_1.html#D7" title="PROB_1:def.7">PROB_1:def 7</a></i>;<br/><a NAME="E3:12_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></i>;<br/><a NAME="E4:12_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D4" title="PROB_1:def.4">PROB_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:12_1_1"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D4" title="PROB_1:def.4">PROB_1:def 4</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>
<b>hence </b><a NAME="E3:12_1"/>
 <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>
 <b>by </b><i><a class="ref" href="prob_1.html#D6" title="PROB_1:def.6">PROB_1:def 6</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>

<b>assume </b><a NAME="E2:12"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</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">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> implies <font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font> )</span><br/><b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font></span><br/><a NAME="E2:12_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="prob_1.html#D6" title="PROB_1:def.6">PROB_1:def 6</a></i>;<br/><a NAME="E3:12_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c2">ASeq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D4" title="PROB_1:def.4">PROB_1:def 4</a></i>;<br/><a NAME="E4:12_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="prob_1.html#D4" title="PROB_1:def.4">PROB_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:12_2"/>
<font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">ASeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</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>
<b>hence </b><a NAME="E4:12"/>
<font color="Maroon" title="c2">ASeq</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>
 <b>by </b><i><a class="ref" href="prob_1.html#D7" title="PROB_1:def.7">PROB_1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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