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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>) <b>-&gt; </b>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">E</font>,<span class="p3">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> $1</span>)</span></span>)</span></span>)</span> =  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:36_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <b>and </b><br/><a NAME="E3:36_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S4" title="FUNCT_1:sch.4">FUNCT_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">f1</font> = <font color="Maroon" title="c4">f</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="osalg_4.html#E2:36_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">f2</font> = <font color="Maroon" title="c5">f1</font>;<br/>
<a NAME="E5:36_1_1"/>
<font color="Maroon" title="c5">f1</font> is <a href="osalg_1.html#V11" title="OSALG_1:attr.11">order-sorted</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">s1</font>, <font color="Maroon" title="c8">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a><br/>

<b>assume </b><a NAME="E1:36_1_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c7">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">s2</font>
 ;<br/>

<a NAME="E2:36_1_1_1"/>
( <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c7">s1</font> &amp; <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c8">s2</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E3:36_1_1"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:36_1_1_1"/>
<font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E1:36_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="osalg_4.html#T11" target="_self" title="OSALG_4:th.11">Th11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:36_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> st <br/>for <font color="Olive" title="b2">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Olive" title="b2">s1</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E3:36_1_1"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>


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