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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">H</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] );<br/>
<b>consider </b><font color="Maroon" title="c1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:3_1"/><i><font color="Green" title="E1">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c1">X</font>
;<br/>

<b>thus </b><a NAME="E3:3_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon">F<sub>1</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:3_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:3_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="group_4.html#E2:3_1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c2">H</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>thus </b><a NAME="E4:3_1"/>
( <font color="Maroon" title="c2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_2"/>
<font color="Maroon" title="c2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:3_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">H1</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H1</font>] )
 
<b>by </b><i><a class="txt" href="group_4.html#E2:3_1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 ;<br/>


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

<b>assume </b><a NAME="E5:3_1"/><i><font color="Green" title="E2">A2</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 ;<br/>

<a NAME="E6:3_1"/>
<font color="Maroon" title="c2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:3_1"/>
<font color="Maroon" title="c2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="group_4.html#E2:3_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="group_4.html#E5:3_1"><i><font color="Green" title="E2">A2</font></i></a></i>;<br/>


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