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<span class="kw">set </span><font color="Maroon" title="c4">G</font> =  <a href="prob_3.html#K6" title="PROB_3:func.6">Complement</a> <font color="Maroon" title="c3">F</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">Complement</a> <font color="Maroon" title="c3">F</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">Complement</a> <font color="Maroon" title="c3">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:56_1_1"/><span class="lab"><font color="Green" title="E45">W1</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">Complement</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:56_1_1"/><span class="lab"><font color="Green" title="E46">W2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">Complement</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1"><span class="lab"><font color="Green" title="E45">W1</font></span></a></span>;<br/>
<a NAME="E6:56_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">Complement</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1_1"><span class="lab"><font color="Green" title="E45">W1</font></span></a>, <a class="ref" href="prob_3.html#D5" target="_self" title="PROB_3:def.5">Def8</a></span>;<br/>
<a NAME="E7:56_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="prob_3.html#K6" title="PROB_3:func.6">Complement</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c7">y</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T20" title="PROB_1:th.20">PROB_1:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:56_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1"><span class="lab"><font color="Green" title="E46">W2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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