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<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="E2:98_1"/><i><font color="Green" title="E50">A3</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> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>1</sub></font>()) &amp; <font color="Maroon">P<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>consider </b><font color="Maroon" title="c2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:98_1"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>]
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:98"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="monoid_0.html#E2:98_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="monoid_0.html#E4:98_1"><i><font color="Green" title="E51">A4</font></i></a></i>;<br/>
<a NAME="E6:98_1"/>
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>1</sub></font>())
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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>thus </b><a NAME="E1:98_1_1"/>
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>1</sub></font>()) )
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:98_1"><i><font color="Green" title="E50">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c3">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<div><i><font color="Green" title="E52">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><a NAME="E1:98_1_2"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">y</font>] )
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:98_1"><i><font color="Green" title="E50">A3</font></i></a></i>;<br/><a NAME="E2:98_1_2"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">y</font>]
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:98"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:98_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:98_1"><i><font color="Green" title="E50">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">H</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E10:98_1"/><i><font color="Green" title="E53">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font>
 <b>from </b><i><a class="ref" href="monoid_0.html#S1" target="_self" title="MONOID_0:sch.1">MONOID_0:sch 1</a>(<a class="txt" href="monoid_0.html#E8:98_1"><i><font color="Green" title="E52">A5</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c5">H</font>
;<br/>

<b>thus </b><a NAME="E11:98_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:98_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="monoid_0.html#E10:98_1"><i><font color="Green" title="E53">A6</font></i></a></i>;<br/>


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