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<b>set </b><font color="Maroon">c<sub>1</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="pboole.html#K9">--&gt;</a> <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E1:39_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pboole.html#K9">--&gt;</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="pboole.html#K9">--&gt;</a> <a href="xboole_0.html#K1">{}</a>  as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:39_1_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E4:39_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/> not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="triang_1.html#E3:39_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E5:39_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="triang_1.html#V1" target="_self">lower_non-empty</a>
 
<b>by </b><i><a class="ref" href="triang_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="triang_1.html#K6" target="_self">NatEmbSeq</a> , <a href="triang_1.html#K4" target="_self">FuncsSeq</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>take </b>
 <a href="triang_1.html#G1" target="_self">TriangStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> #)
;<br/>

<b>thus </b><a NAME="E7:39_1_1"/>
(  <a href="triang_1.html#G1" target="_self">TriangStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) is <a href="triang_1.html#V3" target="_self">lower_non-empty</a> &amp;  <a href="triang_1.html#G1" target="_self">TriangStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) is <a href="triang_1.html#V2" target="_self">strict</a> )
 <b>by </b><i><a class="txt" href="triang_1.html#E5:39_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="triang_1.html#D9" target="_self">Def9</a></i>;<br/>


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