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<b>let </b><font color="Maroon" title="c1">I</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"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font>, <font color="Olive" title="b3">Z</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">X</font> <a href="pboole.html#R7" title="PBOOLE:pred.7">[=</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> <a href="pboole.html#R7" title="PBOOLE:pred.7">[=</a> <font color="Olive" title="b3">Z</font> holds <br/><font color="Olive" title="b1">X</font> <a href="pboole.html#R7" title="PBOOLE:pred.7">[=</a> <font color="Olive" title="b3">Z</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font>, <font color="Maroon" title="c4">Z</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <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="pboole.html#R7" title="PBOOLE:pred.7">[=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R7" title="PBOOLE:pred.7">[=</a> <font color="Maroon" title="c4">Z</font> implies <font color="Maroon" title="c2">X</font> <a href="pboole.html#R7" title="PBOOLE:pred.7">[=</a> <font color="Maroon" title="c4">Z</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:123"/><i><font color="Green" title="E69">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="pboole.html#R7" title="PBOOLE:pred.7">[=</a> <font color="Maroon" title="c3">Y</font>
 <b>and </b><br/><a NAME="E2:123"/><i><font color="Green" title="E70">A2</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="pboole.html#R7" title="PBOOLE:pred.7">[=</a> <font color="Maroon" title="c4">Z</font>
 ; <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="pboole.html#R7" title="PBOOLE:pred.7">[=</a> <font color="Maroon" title="c4">Z</font></span><br/>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D14" target="_self" title="PBOOLE:def.14">PBOOLE:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">X</font> implies <font color="Maroon" title="c5">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c4">Z</font> )</span><br/>

<b>assume </b><a NAME="E3:123"/>
<font color="Maroon" title="c5">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c4">Z</font></span><br/>

<a NAME="E4:123"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="txt" href="pboole.html#E1:123"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="pboole.html#D14" target="_self" title="PBOOLE:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E5:123"/>
<font color="Maroon" title="c5">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="txt" href="pboole.html#E2:123"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="pboole.html#D14" target="_self" title="PBOOLE:def.14">Def14</a></i>; <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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