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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">X</font>, <font color="Olive" title="b3">x</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> holds <br/><font color="Olive" title="b1">y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">x</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</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">y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> implies <font color="Maroon" title="c2">y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</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">y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">i</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="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:54"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</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">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

<a NAME="E3:54"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a></span>;<br/>
<a NAME="E4:54"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="pboole.html#D9" title="PBOOLE:def.9">PBOOLE:def 9</a></span>;<br/>
<a NAME="E5:54"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="pzfmisc1.html#D1" target="_self" title="PZFMISC1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:54"/>
<font color="Maroon" title="c2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</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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