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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b1">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="mathmorp.html#K6" title="MATHMORP:func.6">(o)</a> <font color="Olive" title="b2">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K6" title="MATHMORP:func.6">(o)</a> <font color="Maroon" title="c3">Y</font> )</span><br/>





<a NAME="E1:59"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K6" title="MATHMORP:func.6">(o)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">Y</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E2:59"/>
<font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mathmorp.html#T50" target="_self" title="MATHMORP:th.50">Th50</a></span>; <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="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K6" title="MATHMORP:func.6">(o)</a> <font color="Maroon" title="c3">Y</font></span><br/>

<a NAME="E3:59"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K6" title="MATHMORP:func.6">(o)</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">Y</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:59"/>
<font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="mathmorp.html#K6" title="MATHMORP:func.6">(o)</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mathmorp.html#T50" target="_self" title="MATHMORP:th.50">Th50</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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