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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> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b1">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Olive" title="b2">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> iff <font color="Olive" title="b1">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">Y</font> )</span><br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</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#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> iff <font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font> )</span><br/>





<div><b>hereby </b> <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#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:58_1"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <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#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/><b>assume </b><a NAME="E1:58_1_1"/>
<font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="ref" href="pboole.html#D9" target="_self" title="PBOOLE:def.9">Def9</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="#E1:58_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="pboole.html#T5" target="_self" title="PBOOLE:th.5">Th5</a></i>
;<br/><b>hence </b><a NAME="E3:58_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:58_1"/>
<font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5" target="_self" title="PBOOLE:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:58"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</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#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><b>assume </b><a NAME="E1:58_2"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><a NAME="E2:58_2"/><b>then </b><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="pboole.html#D5" target="_self" title="PBOOLE:def.5">Def5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:58_2"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="pboole.html#D9" target="_self" title="PBOOLE:def.9">Def9</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="#E2:58_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
<font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#T6" target="_self" title="PBOOLE:th.6">Th6</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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