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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="Red">::  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> 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="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#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">PA</font>,<font color="Olive" title="b1">G</font> <a href="bvfunc_1.html#R2" title="BVFUNC_1:pred.2">is_dependent_of</a>  <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Olive" title="b3">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="partit1.html#K1" title="PARTIT1: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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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="b2">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#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">PA</font>,<font color="Maroon" title="c2">G</font> <a href="bvfunc_1.html#R2" title="BVFUNC_1:pred.2">is_dependent_of</a>  <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Olive" title="b2">PA</font>,<font color="Maroon" title="c2">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="Red">::  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#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> <a href="bvfunc_1.html#R2" title="BVFUNC_1:pred.2">is_dependent_of</a>  <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <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="c4">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="Red">::  thesis: </font></span></a><span class="hide">  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font> <a href="bvfunc_1.html#R2" title="BVFUNC_1:pred.2">is_dependent_of</a>  <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="bvfunc_1.html#D18" title="BVFUNC_1:def.18">BVFUNC_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font> or for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> or <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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> or <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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:19"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> holds <br/><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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> implies <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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c9">x2</font> = <font color="Maroon" title="c7">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/>
<a NAME="E3:19_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c9">x2</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="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></span>;<br/>
<a NAME="E4:19_1"/>
( <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> or <font color="Maroon" title="c5">F</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<a NAME="E5:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A4</font></span>: 
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c8">x1</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E3:19_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_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="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="c8">x1</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp;  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="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">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="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="c8">x1</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp; ( 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="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">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="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="c8">x1</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) ) &amp;  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="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">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="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="c8">x1</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) ) &amp; ( 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="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
(  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="c8">x1</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp;  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="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span><br/><div class="add"><a NAME="E2:19_1_1_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> <font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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="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="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1_1"/>
<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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="19_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_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="c8">x1</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp; ( 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="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:19_1_1_2"/>
<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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="19_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_3"/>
( ( 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="c8">x1</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) ) &amp;  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="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:19_1_1_3"/>
<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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="19_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_4"/><span class="lab"><font color="Green" title="E11">A6</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>  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="c8">x1</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) ) &amp; ( 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="c9">x2</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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#K8" title="MARGREL1:func.8">TRUE</a>  ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span><br/><div class="add"><a NAME="E2:19_1_1_4"/><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> <font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">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="c8">x1</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#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1_4"/>
<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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_4"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>


</div>
