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<div><span class="kw">hereby </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="roughs_1.html#R4" title="ROUGHS_1:pred.4">_=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R5" title="ROUGHS_1:pred.5">=^</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R6" title="ROUGHS_1:pred.6">_=^</a> <font color="Maroon" title="c3">Y</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:94_3_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R6" title="ROUGHS_1:pred.6">_=^</a> <font color="Maroon" title="c3">Y</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="c2">X</font> <a href="roughs_1.html#R4" title="ROUGHS_1:pred.4">_=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R5" title="ROUGHS_1:pred.5">=^</a> <font color="Maroon" title="c3">Y</font> )</span><br/><a NAME="E2:94_3_1_1"/><span class="kw">then </span>
(  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c3">Y</font> &amp;  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c3">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_1.html#D16" target="_self" title="ROUGHS_1:def.16">Def16</a></span>;<br/><span class="kw">hence </span><a NAME="E3:94_3_1_1"/>
( <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R4" title="ROUGHS_1:pred.4">_=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R5" title="ROUGHS_1:pred.5">=^</a> <font color="Maroon" title="c3">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_1.html#D14" target="_self" title="ROUGHS_1:def.14">Def14</a>, <a class="ref" href="roughs_1.html#D15" target="_self" title="ROUGHS_1:def.15">Def15</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:94_3_1"/>
( <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R4" title="ROUGHS_1:pred.4">_=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R5" title="ROUGHS_1:pred.5">=^</a> <font color="Maroon" title="c3">Y</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="c2">X</font> <a href="roughs_1.html#R6" title="ROUGHS_1:pred.6">_=^</a> <font color="Maroon" title="c3">Y</font></span><br/>

<a NAME="E3:94_3_1"/><span class="kw">then </span>
(  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <font color="Maroon" title="c3">Y</font> &amp;  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c3">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_1.html#D14" target="_self" title="ROUGHS_1:def.14">Def14</a>, <a class="ref" href="roughs_1.html#D15" target="_self" title="ROUGHS_1:def.15">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:94_3_1"/>
<font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R6" title="ROUGHS_1:pred.6">_=^</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_1.html#D16" target="_self" title="ROUGHS_1:def.16">Def16</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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