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<b>set </b><font color="Maroon" title="c1">Sk</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<a NAME="E1:39_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">Sk</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:39_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> holds <br/>for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/> not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> holds <br/> not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>

<a NAME="E1:39_1_1_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:39_1_1_1"/>
not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> holds <br/> not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/>

<b>hence </b><a NAME="E3:39_1_1_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> holds <br/> not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="#E1:39_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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>
<a NAME="E4:39_1_1"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c2">Sk</font> is <a href="triang_1.html#V1" title="TRIANG_1:attr.1">lower_non-empty</a>
 
<b>by </b><i><a class="ref" href="triang_1.html#D4" target="_self" title="TRIANG_1:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">A</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of  <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a> , <a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> <font color="Maroon" title="c2">Sk</font>;<br/>
<b>take </b>
 <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c2">Sk</font>,<font color="Maroon" title="c3">A</font> #)
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c2">Sk</font>,<font color="Maroon" title="c3">A</font> #) is <a href="triang_1.html#V3" title="TRIANG_1:attr.3">lower_non-empty</a> &amp;  <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c2">Sk</font>,<font color="Maroon" title="c3">A</font> #) is <a href="triang_1.html#V2" title="TRIANG_1:attr.2">strict</a> )</span><br/>

<b>thus </b><a NAME="E6:39_1_1"/>
(  <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c2">Sk</font>,<font color="Maroon" title="c3">A</font> #) is <a href="triang_1.html#V3" title="TRIANG_1:attr.3">lower_non-empty</a> &amp;  <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c2">Sk</font>,<font color="Maroon" title="c3">A</font> #) is <a href="triang_1.html#V2" title="TRIANG_1:attr.2">strict</a> )
 <b>by </b><i><a class="txt" href="#E4:39_1_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="triang_1.html#D9" target="_self" title="TRIANG_1:def.9">Def9</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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