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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  holds  <font color="Olive" title="b1">a</font> <a href="bvfunc_1.html#R2" title="BVFUNC_1:pred.2">is_dependent_of</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font> be  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#R2" title="BVFUNC_1:pred.2">is_dependent_of</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span><br/>



<a NAME="E1:48"/>
for <font color="Olive" title="b1">F</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font> holds <br/>for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">F</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font> implies 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="c3">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="c3">F</font> holds <br/><font color="Maroon" title="c2">a</font> <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> <font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> )</span><br/>

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c3">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="c3">F</font> holds <br/><font color="Maroon" title="c2">a</font> <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> <font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font></span><br/>

<a NAME="E2:48_1"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x2</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">O_el</a> <font color="Maroon" title="c1">Y</font> or <font color="Maroon" title="c2">a</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">I_el</a> <font color="Maroon" title="c1">Y</font> )
 </span><b>by </b><i><a class="ref" href="bvfunc_1.html#T24" target="_self" title="BVFUNC_1:th.24">Th24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">O_el</a> <font color="Maroon" title="c1">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x2</font></span><br/><div class="add"><a NAME="E2:48_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def13</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_1_1_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_2"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">I_el</a> <font color="Maroon" title="c1">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x2</font></span><br/><div class="add"><a NAME="E2:48_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_1_1_1_2"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48_1"/>
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="c3">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="c3">F</font> holds <br/><font color="Maroon" title="c2">a</font> <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> <font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_1"><i><font color="Green" title="E24">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:48"/>
<font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#R2" title="BVFUNC_1:pred.2">is_dependent_of</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D18" target="_self" title="BVFUNC_1:def.18">Def18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
