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<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="finset_1.html#V2" title="FINSET_1:attr.2">V19</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E11">A1</font></i>: 
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>  st <font color="Olive" title="b1">X</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; not <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> holds <br/><font color="Olive" title="b2">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">X</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<a NAME="E3:47"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a>, <a class="ref" href="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a></i>;<br/>
<a NAME="E4:47"/>
for <font color="Olive" title="b1">X'</font>, <font color="Olive" title="b2">Y'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Olive" title="b2">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> &amp; not <font color="Olive" title="b1">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y'</font> holds <br/><font color="Olive" title="b2">Y'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">X'</font>, <font color="Maroon" title="c5">Y'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c4">X'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c5">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )
 ;<br/>

<b>assume </b><a NAME="E2:47_1"/><i><font color="Green" title="E15">A5</font></i>: 
not <font color="Maroon" title="c4">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y'</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">M</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E4:47_1"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#T146" title="PBOOLE:th.146">PBOOLE:146</a></i>;<br/>
<a NAME="E5:47_1"/><i><font color="Green" title="E17">A7</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c5">Y'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:47_1"/><i><font color="Green" title="E18">A8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c5">Y'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="mboolean.html#E2:47"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="mboolean.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<a NAME="E7:47_1"/><i><font color="Green" title="E19">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">X'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E8:47_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">X'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="mboolean.html#E2:47"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="mboolean.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">K1</font> = <font color="Maroon" title="c6">M</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">X'</font></span>)</span>, <font color="Maroon" title="c8">K2</font> = <font color="Maroon" title="c6">M</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c5">Y'</font></span>)</span> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="mboolean.html#E6:47_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:47_1"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E21">A11</font></i>: <font color="Maroon" title="c7">K1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">X'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="mboolean.html#E7:47_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c4">X'</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/>
<i><font color="Green" title="E22">A12</font></i>: <font color="Maroon" title="c8">K2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c5">Y'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="mboolean.html#E5:47_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="mboolean.html#E10:47_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c5">Y'</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/>
<a NAME="E13:47_1"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c7">K1</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:47_1_3"/><i><font color="Green" title="E24">A14</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_3_1_1"/>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> or <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3_1_1_1"><b>case </b></a><a NAME="E1:47_1_3_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_3_1_1_1"/>
<font color="Maroon" title="c7">K1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="mboolean.html#E1:47_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="mboolean.html#E11:47_1"><i><font color="Green" title="E21">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3_1_1_2"><b>case </b></a><a NAME="E1:47_1_3_1_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><a NAME="E2:47_1_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">X'</font></span>)</span>
 <b>by </b><i><a class="txt" href="mboolean.html#E7:47_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:47_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">K1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:47_1_3_1_1_2"/>
<font color="Maroon" title="c7">K1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="mboolean.html#E4:47_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="mboolean.html#E1:47_1_3"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1_3"/>
<font color="Maroon" title="c7">K1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:47_1"/>
<font color="Maroon" title="c8">K2</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:47_1_4"/><i><font color="Green" title="E24">A15</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_4_1_1"/>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> or <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_4_1_1_1"><b>case </b></a><a NAME="E1:47_1_4_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_4_1_1_1"/>
<font color="Maroon" title="c8">K2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="mboolean.html#E1:47_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="mboolean.html#E12:47_1"><i><font color="Green" title="E22">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_4_1_1_2"><b>case </b></a><a NAME="E1:47_1_4_1_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><a NAME="E2:47_1_4_1_1_2"/><b>then </b>
not <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c5">Y'</font></span>)</span>
 <b>by </b><i><a class="txt" href="mboolean.html#E5:47_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:47_1_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">K2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:47_1_4_1_1_2"/>
<font color="Maroon" title="c8">K2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="mboolean.html#E4:47_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="mboolean.html#E1:47_1_4"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1_4"/>
<font color="Maroon" title="c8">K2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:47_1"/><b>then </b>
( <font color="Maroon" title="c7">K1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c8">K2</font> or <font color="Maroon" title="c8">K2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">K1</font> )
 
<b>by </b><i><a class="txt" href="mboolean.html#E1:47"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="mboolean.html#E13:47_1"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:47_1"/>
<font color="Maroon" title="c5">Y'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X'</font>
 <b>by </b><i><a class="txt" href="mboolean.html#E2:47"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="mboolean.html#E2:47_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="mboolean.html#E11:47_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="mboolean.html#E12:47_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 
<b>by </b><i><a class="txt" href="mboolean.html#E3:47"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="card_2.html#T81" title="CARD_2:th.81">CARD_2:81</a></i>;<br/>
<b>hence </b><a NAME="E6:47"/>
<span class="p1">(<span class="default"><a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="mboolean.html#E2:47"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="mboolean.html#D2" target="_self" title="MBOOLEAN:def.2">Def2</a></i>;<br/>


</div>
