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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>] or <font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:17_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:17_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:17_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="scheme1.html#E2:17_1"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>) ) &amp; ( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>] implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a<sub>1</sub></font>) ) &amp; ( <font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">a<sub>1</sub></font>] implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a<sub>1</sub></font>) ) );<br/>
<a NAME="E4:17_1"/><i><font color="Green">E10</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> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:17_1_2"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="scheme1.html#E2:17_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_2_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>] or <font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 </span><b>by </b><i><a class="txt" href="scheme1.html#E2:17_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="scheme1.html#E1:17_1_2"><i><font color="Green">E11</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_2_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>);<br/><a NAME="E2:17_1_2_1_1_1"/>
( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; <font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 <b>by </b><i><a class="txt" href="scheme1.html#E1:17"><i><font color="Green">E4</font></i></a>, <a class="txt" href="scheme1.html#E2:17_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="scheme1.html#E1:17_1_2_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>hence </b><a NAME="E3:17_1_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_2_1_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>2</sub></font>);<br/><a NAME="E2:17_1_2_1_1_2"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; <font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 <b>by </b><i><a class="txt" href="scheme1.html#E1:17"><i><font color="Green">E4</font></i></a>, <a class="txt" href="scheme1.html#E2:17_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="scheme1.html#E1:17_1_2_1_1_2"><i><font color="Green">E13</font></i></a></i>;<br/><b>hence </b><a NAME="E3:17_1_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_2_1_1_3"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>2</sub></font>);<br/><a NAME="E2:17_1_2_1_1_3"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 <b>by </b><i><a class="txt" href="scheme1.html#E1:17"><i><font color="Green">E4</font></i></a>, <a class="txt" href="scheme1.html#E2:17_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="scheme1.html#E1:17_1_2_1_1_3"><i><font color="Green">E13</font></i></a></i>;<br/><b>hence </b><a NAME="E3:17_1_2_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:17_1"/><i><font color="Green">E11</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( 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> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="scheme1.html#E4:17_1"><i><font color="Green">E10</font></i></a>);<br/></i>
<a NAME="E7:17_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_1_3"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_3_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>] or <font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c<sub>4</sub></font>] )
 </span><b>by </b><i><a class="txt" href="scheme1.html#E2:17_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1_3"><i><font color="Green">E12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_3_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><div class="add"><a NAME="E2:17_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1_3"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:17_1_3_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="scheme1.html#E2:17"><i><font color="Green">E5</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="scheme1.html#E1:17_1_3_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_3_1_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><div class="add"><a NAME="E2:17_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1_3"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:17_1_3_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="scheme1.html#E3:17"><i><font color="Green">E6</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="scheme1.html#E1:17_1_3_1_1_2"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_3_1_1_3"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><div class="add"><a NAME="E2:17_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1_3"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:17_1_3_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="scheme1.html#E4:17"><i><font color="Green">E7</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scheme1.html#E3:17_1_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="scheme1.html#E1:17_1_3_1_1_3"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:17_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="scheme1.html#E3:17_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E9:17_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] or <font color="Maroon">P<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) ) )
 <b>by </b><i><a class="txt" href="scheme1.html#E2:17_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E10:17_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>hence </b><a NAME="E11:17_1"/>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>] implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>) ) &amp; ( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>] implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>4</sub></font>) ) &amp; ( <font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c<sub>4</sub></font>] implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>) ) )
 <b>by </b><i><a class="txt" href="scheme1.html#E6:17_1"><i><font color="Green">E11</font></i></a></i>;<br/>


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