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<span class="kw">let </span><font color="Maroon" title="c1">SFX</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="setfam_1.html#K4" title="SETFAM_1:func.4">DIFFERENCE</a> <font color="Maroon" title="c1">SFX</font>,<font color="Maroon" title="c1">SFX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c1">SFX</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="setfam_1.html#D2" target="_self" title="SETFAM_1:def.2">SETFAM_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K4" title="SETFAM_1:func.4">DIFFERENCE</a> <font color="Maroon" title="c1">SFX</font>,<font color="Maroon" title="c1">SFX</font> implies  ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">SFX</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:31"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K4" title="SETFAM_1:func.4">DIFFERENCE</a> <font color="Maroon" title="c1">SFX</font>,<font color="Maroon" title="c1">SFX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">SFX</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">Z1</font>, <font color="Maroon" title="c4">Z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:31"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">SFX</font>
 <span class="kw">and </span><br/><a NAME="E4:31"/>
<font color="Maroon" title="c4">Z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">SFX</font>
 <span class="kw">and </span><br/><a NAME="E5:31"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Z1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">Z2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D6" target="_self" title="SETFAM_1:def.6">Def6</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">Z1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">SFX</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E6:31"/>
( <font color="Maroon" title="c3">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">SFX</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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