<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><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/>
<span class="kw">reconsider </span><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#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">A</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  <a href="triang_1.html#K4" title="TRIANG_1:func.4">NatEmbSeq</a> , <a href="triang_1.html#K2" title="TRIANG_1:func.2">FuncsSeq</a> <font color="Maroon" title="c2">Sk</font>;<br/>
<span class="kw">take </span>
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<a NAME="E3:27_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="27_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c4">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="c4">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/>

<span class="kw">assume </span><a NAME="E1:27_1_1_1"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">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="c4">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="E2:27_1_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:27_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="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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>
<a NAME="E4:27_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">Sk</font> is  <a href="triang_1.html#V1" title="TRIANG_1:attr.1">lower_non-empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="triang_1.html#D3" target="_self" title="TRIANG_1:def.3">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:27_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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="triang_1.html#D8" target="_self" title="TRIANG_1:def.8">Def9</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>
