<?xml version="1.0"?>
<div class="add">

<a NAME="E1:9_1_1"/>
<font color="Maroon" title="c2">M</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:9_1_1"/>
not  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is  <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>  &amp;  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is  <a href="card_3.html#V2" title="CARD_3:attr.2">with_common_domain</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:9_1_1"/>
 <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is  <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> is  <a href="card_3.html#V2" title="CARD_3:attr.2">with_common_domain</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</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="funct_1.html#D19" title="FUNCT_1:def.19">FUNCT_1:def 19</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> or <font color="Maroon" title="c3">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font>
 ; <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="c3">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:9_1_1_1"/>
<font color="Maroon" title="c3">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</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>

<span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="card_3.html#D10" title="CARD_3:def.10">CARD_3:def 10</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="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> or  not <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font> or  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">g</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:9_1_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E5:9_1_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K1" title="CLOSURE2:func.1">Bool</a> <font color="Maroon" title="c2">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">g</font></span><br/>

<a NAME="E6:9_1_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c4">g</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></span>;<br/>
<a NAME="E7:9_1_1"/>
<font color="Maroon" title="c3">f</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> = 
<font color="Maroon" title="c1">I</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
;<br/> <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>
