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<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font> implies  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">z</font>] )</span><br/>

<b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="ref" href="partit1.html#D3" target="_self" title="PARTIT1:def.3">Def3</a></i>;<br/>
<b>take </b>
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>, <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">x</font>]</span><br/>

<b>thus </b><a NAME="E3:28_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>, <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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