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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#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">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="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span><br/>  for <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>)<br/>  for <font color="Olive" title="b5">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  <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="Olive" title="b2">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Olive" title="b3">c</font></span>)</span>,<font color="Olive" title="b5">PA</font>,<font color="Olive" title="b1">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Olive" title="b4">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span>,<font color="Olive" title="b5">PA</font>,<font color="Olive" title="b1">G</font>)</span>)</span> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Olive" title="b2">b</font></span>)</span></span>)</span>,<font color="Olive" title="b5">PA</font>,<font color="Olive" title="b1">G</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="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>; <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">b</font>, <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">a</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">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  <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="Olive" title="b1">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Olive" title="b2">c</font></span>)</span>,<font color="Olive" title="b4">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span>,<font color="Olive" title="b4">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span>,<font color="Olive" title="b4">PA</font>,<font color="Maroon" title="c2">G</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">a</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</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">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  <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">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">c</font></span>)</span>,<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> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<font color="Olive" title="b1">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<font color="Olive" title="b1">PA</font>,<font color="Maroon" title="c2">G</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c7">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" title="BVFUNC_1:def.12">BVFUNC_1:def 12</a> <a class="txt" 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"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p4">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  or <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )</span><br/>













<span class="kw">assume </span><a NAME="E1:39"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p4">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/>

<a NAME="E2:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</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> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p4">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D20" title="MARGREL1:def.20">MARGREL1:def 20</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1"/>
 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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> (<font color="Maroon" title="c7">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="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) or  not <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <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#K8" title="MARGREL1:func.8">TRUE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:39_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D17" title="BVFUNC_1:def.17">BVFUNC_1:def 17</a></span>;<br/><a NAME="E3:39_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> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_2.html#D10" title="BVFUNC_2:def.10">BVFUNC_2:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="margrel1.html#T12" title="MARGREL1:th.12">MARGREL1:12</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">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E5:39"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> (<font color="Maroon" title="c7">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="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E6:39"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/>
<a NAME="E7:39"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="margrel1.html#D20" title="MARGREL1:def.20">MARGREL1:def 20</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> (<font color="Maroon" title="c7">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="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>) &amp;  not <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">c</font></span>)</span> <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#K8" title="MARGREL1:func.8">TRUE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:39_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">B_INF</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D16" title="BVFUNC_1:def.16">BVFUNC_1:def 16</a></span>;<br/><a NAME="E3:39_2"/><span class="kw">then </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">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_2.html#D9" title="BVFUNC_2:def.9">BVFUNC_2:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="margrel1.html#T12" title="MARGREL1:th.12">MARGREL1:12</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>
<a NAME="E9:39"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<a NAME="E10:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D8" title="BVFUNC_1:def.8">BVFUNC_1:def 8</a></span>;<br/>
<a NAME="E11:39"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
(  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  or  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3"><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="E1:39_3"/>
( ( <font color="Maroon" title="c5">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp;  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <font color="Maroon" title="c4">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="binarith.html#T3" title="BINARITH:th.3">BINARITH:3</a>, <a class="ref" href="margrel1.html#T12" title="MARGREL1:th.12">MARGREL1:12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3_1"><span class="kw">suppose </span></a><a NAME="E1:39_3_1"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp;  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D20" title="MARGREL1:def.20">MARGREL1:def 20</a></span>
<br/>.= 
<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_3_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/><a NAME="E3:39_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> (<font color="Maroon" title="c6">PA</font>,<font color="Maroon" title="c2">G</font>)</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="bvfunc_1.html#D17" title="BVFUNC_1:def.17">BVFUNC_1:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_3_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_2.html#D10" title="BVFUNC_2:def.10">BVFUNC_2:def 10</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="39_3_2"><span class="kw">suppose </span></a><a NAME="E1:39_3_2"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <font color="Maroon" title="c4">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/><div class="add"><a NAME="E2:39_3_2"/><span class="kw">then </span>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_3_2"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">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="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_3_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="margrel1.html#T11" title="MARGREL1:th.11">MARGREL1:11</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>

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