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<span class="kw">let </span><font color="Maroon" title="c1">Y</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span><br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">u</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a><br/>  for <font color="Olive" title="b4">PA</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Olive" title="b3">u</font></span>)</span>,<font color="Olive" title="b4">PA</font>,<font color="Olive" title="b1">G</font>) <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">PA</font>,<font color="Olive" title="b1">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Olive" title="b3">u</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">u</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a><br/>  for <font color="Olive" title="b3">PA</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Olive" title="b2">u</font></span>)</span>,<font color="Olive" title="b3">PA</font>,<font color="Maroon" title="c2">G</font>) <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Olive" title="b2">u</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">u</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">PA</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span>,<font color="Olive" title="b1">PA</font>,<font color="Maroon" title="c2">G</font>) <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">PA</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>) <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="bvfunc_1.html#D12" onmouseover="rs('bvfunc_1/D12')" onmouseout="rh()">BVFUNC_1:def 12</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )</span><br/>











<a NAME="E1:38"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D6" onmouseover="rs('eqrel_1/D6')" onmouseout="rh()">EQREL_1:def 6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:38"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> </span><br/>

<a NAME="E3:38"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) holds <br/>( <font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  &amp; <font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) implies ( <font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  &amp; <font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  &amp; <font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )</span><br/>

<a NAME="E2:38_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" onmouseover="rs('bvfunc_1/D16')" onmouseout="rh()">BVFUNC_1:def 16</a></span>;<br/>
<a NAME="E3:38_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D20" onmouseover="rs('margrel1/D20')" onmouseout="rh()">MARGREL1:def 20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:38_1"/>
( <font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  &amp; <font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T12" onmouseover="rs('margrel1/T12')" onmouseout="rh()">MARGREL1:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E4:38"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D20" onmouseover="rs('margrel1/D20')" onmouseout="rh()">MARGREL1:def 20</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_2"/>
(  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) holds <br/><font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  or  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) &amp;  not <font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1"><span class="kw">suppose </span></a><a NAME="E1:38_2_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) holds <br/><font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> </span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_2_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_2_1_1_1"/>
(  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) &amp; <font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  ) or  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) or  not <font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) &amp; <font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> </span><br/><div class="add"><a NAME="E2:38_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D17" onmouseover="rs('bvfunc_1/D17')" onmouseout="rh()">BVFUNC_1:def 17</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> = 
<a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E1:38_2_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="eqrel_1.html#D6" onmouseover="rs('eqrel_1/D6')" onmouseout="rh()">EQREL_1:def 6</a></span>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 

;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="38_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_2_1_1_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) or  not <font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> </span><br/><div class="add"><a NAME="E2:38_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D6" onmouseover="rs('eqrel_1/D6')" onmouseout="rh()">EQREL_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><span class="kw">hence </span><a NAME="E3:38_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="38_2_2"><span class="kw">suppose </span></a><a NAME="E1:38_2_2"/>
 ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">EqClass</a> (<font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) &amp;  not <font color="Maroon" title="c4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:38_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
