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<div class="add">

<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/>  for <font color="Olive" title="b2">X</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st (  for <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> holds <br/> ( <font color="Olive" title="b3">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">X</font> iff <font color="Olive" title="b3">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">Y</font> ) ) holds <br/><font color="Olive" title="b2">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st (  for <font color="Olive" title="b2">x</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b2">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">X</font> iff <font color="Olive" title="b2">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font> ) ) holds <br/><font color="Olive" title="b1">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</a>() <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="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font> iff <font color="Olive" title="b1">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font> ) ) implies <font color="Maroon" title="c3">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">Y</font> )</span><br/>





<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E1:141"/><span class="lab"><font color="Green" title="E76">A1</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D13" target="_self" title="PBOOLE:def.13">Def16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E3:141"/><span class="lab"><font color="Green" title="E77">A2</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S1" target="_self" title="PBOOLE:sch.1">PBOOLE:sch 1</a>(<a class="txt" href="#E1:141"><span class="lab"><font color="Green" title="E76">A1</font></span></a>);<br/></span>
<span class="kw">assume </span><a NAME="E4:141"/><span class="lab"><font color="Green" title="E78">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font> iff <font color="Olive" title="b1">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">Y</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )</span><br/><span class="kw">assume </span><a NAME="E1:141_1"/><span class="lab"><font color="Green" title="E79">A4</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="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">e</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">e</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> implies <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> ) &amp; ( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> implies <font color="Maroon" title="c6">e</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">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> (<font color="Maroon" title="c5">i</font>,$1,<font color="Maroon" title="c6">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>);<br/><span class="kw">consider </span><font color="Maroon" title="c7">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:141_1_1"/><span class="lab"><font color="Green" title="E80">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <span class="kw">and </span><br/><a NAME="E3:141_1_1"/><span class="lab"><font color="Green" title="E81">A6</font></span>: 
 for <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">u</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">g</font> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:141_1_1"><span class="lab"><font color="Green" title="E80">A5</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><span class="lab"><font color="Green" title="E82">A7</font></span>: <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <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="#E1:141_1"><span class="lab"><font color="Green" title="E79">A4</font></span></a>, <a class="txt" href="#E3:141_1_1"><span class="lab"><font color="Green" title="E81">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> implies <font color="Maroon" title="c6">e</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>
<div class="add"><span class="kw">assume </span><a NAME="E1:141_1_1_2"/><span class="lab"><font color="Green" title="E83">A8</font></span>: 
<font color="Maroon" title="c6">e</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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></span><br/><a NAME="E2:141_1_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D1" target="_self" title="PBOOLE:def.1">PBOOLE:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</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="c9">u</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:141_1_1_2_1"/><span class="lab"><font color="Green" title="E84">A9</font></span>: 
<font color="Maroon" title="c9">u</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</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="c9">u</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:141_1_1_2_1_1"/>
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:141_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</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="c9">u</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:141_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</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="c9">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:141_1_1"><span class="lab"><font color="Green" title="E82">A7</font></span></a>, <a class="txt" href="#E1:141_1_1_2"><span class="lab"><font color="Green" title="E83">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:141_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E85">A10</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</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="c9">u</font></span><br/><div class="add"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c6">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:141_1_1"><span class="lab"><font color="Green" title="E81">A6</font></span></a>, <a class="txt" href="#E1:141_1_1_2_1"><span class="lab"><font color="Green" title="E84">A9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:141_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A10</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:141_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</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="c9">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:141"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="txt" href="#E1:141_1_1_2_1"><span class="lab"><font color="Green" title="E84">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:141_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">g</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:141"><span class="lab"><font color="Green" title="E78">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:141_1_1_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:141_1"><span class="lab"><font color="Green" title="E79">A4</font></span></a>, <a class="txt" href="#E5:141_1_1"><span class="lab"><font color="Green" title="E82">A7</font></span></a>, <a class="ref" href="pboole.html#D1" target="_self" title="PBOOLE:def.1">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E7:141_1_1"/><span class="lab"><font color="Green" title="E83">A11</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">e</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="E8:141_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D1" target="_self" title="PBOOLE:def.1">PBOOLE:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:141_1_1_3"/><span class="lab"><font color="Green" title="E84">A12</font></span>: 
<font color="Maroon" title="c9">u</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:141_1_1_3_1"/>
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:141_1_1_3_1_1"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:141_1_1_3_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:141_1_1"><span class="lab"><font color="Green" title="E82">A7</font></span></a>, <a class="txt" href="#E7:141_1_1"><span class="lab"><font color="Green" title="E83">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:141_1_1_3_1_2"/><span class="lab"><font color="Green" title="E85">A13</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font></span><br/><div class="add"><a NAME="E2:141_1_1_3_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:141"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="ref" href="pboole.html#D1" target="_self" title="PBOOLE:def.1">Def4</a></span>;<br/><a NAME="E3:141_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A14</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:141"><span class="lab"><font color="Green" title="E78">A3</font></span></a></span>;<br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c6">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:141_1_1"><span class="lab"><font color="Green" title="E81">A6</font></span></a>, <a class="txt" href="#E1:141_1_1_3"><span class="lab"><font color="Green" title="E84">A12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:141_1_1_3_1_2"><span class="lab"><font color="Green" title="E85">A13</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:141_1_1_3_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:141_1_1_3"><span class="lab"><font color="Green" title="E84">A12</font></span></a>, <a class="txt" href="#E3:141_1_1_3_1_2"><span class="lab"><font color="Green" title="E86">A14</font></span></a>, <a class="ref" href="pboole.html#D1" target="_self" title="PBOOLE:def.1">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E9:141_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">g</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:141"><span class="lab"><font color="Green" title="E78">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:141_1_1"/>
<font color="Maroon" title="c6">e</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="txt" href="#E1:141_1"><span class="lab"><font color="Green" title="E79">A4</font></span></a>, <a class="txt" href="#E5:141_1_1"><span class="lab"><font color="Green" title="E82">A7</font></span></a>, <a class="ref" href="pboole.html#D1" target="_self" title="PBOOLE:def.1">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:141_1"/>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:141"/>
<font color="Maroon" title="c3">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" target="_self" title="PBOOLE:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
