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<b>let </b><font color="Maroon" title="c5">X</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">I</font>;<br/>

<b>thus </b><a NAME="E1:9_1_1"/>
( <font color="Maroon" title="c5">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font> implies for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> holds  <font color="Maroon" title="c5">X</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pralg_1.html#K10" title="PRALG_1:func.10">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font>;<br/>

<a NAME="E2:9_1_1_1"/>
 ex <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="pralg_1.html#K10" title="PRALG_1:func.10">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c5">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</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="Olive" title="b1">P</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:9_1_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:9_1_1_1"/>
<font color="Maroon" title="c5">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pralg_1.html#K10" title="PRALG_1:func.10">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:9_1_1"/><i><font color="Green" title="E3">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> holds  <font color="Maroon" title="c5">X</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pralg_1.html#K10" title="PRALG_1:func.10">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/>

<a NAME="E3:9_1_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> holds <br/> ex <font color="Olive" title="b2">P</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">X</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1_1_2"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">C</font> <a href="pralg_1.html#K10" title="PRALG_1:func.10">.</a> <font color="Maroon" title="c7">i</font>
;<br/>

<b>thus </b><a NAME="E3:9_1_1_2"/>
( <font color="Maroon" title="c4">C</font> <a href="pralg_1.html#K10" title="PRALG_1:func.10">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c5">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pralg_1.html#K10" title="PRALG_1:func.10">.</a> <font color="Maroon" title="c7">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:9_1_1"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:9_1_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>


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