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

<div><i><font color="Green" title="E12">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:17_1_1"/><i><font color="Green" title="E14">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E4:17_1_1"/><i><font color="Green" title="E15">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="partfun1.html#E5"><i><font color="Green" title="E4">Lm2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c2">f</font> = <font color="Maroon" title="c1">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="partfun1.html#E3:17_1_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="partfun1.html#E4:17_1_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="partfun1.html#E6"><i><font color="Green" title="E5">Lm3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">f</font>;<br/><a NAME="E6:17_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="partfun1.html#E1:17_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="partfun1.html#E6"><i><font color="Green" title="E5">Lm3</font></i></a></i>;<br/><a NAME="E7:17_1_1"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E8:17_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>by </b><i><a class="txt" href="partfun1.html#E1:17"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="partfun1.html#E1:17_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/><b>thus </b><a NAME="E9:17_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>by </b><i><a class="txt" href="partfun1.html#E7:17_1_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_2"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c1">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( (  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b1">y</font>] implies <font color="Maroon">P<sub>1</sub></font>[$1,$2] ) &amp; ( ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b1">y</font>] ) implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">y1</font> ) );<br/><a NAME="E3:17_1_2"/><i><font color="Green" title="E13">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> ex <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<a NAME="E2:17_1_2_1"/>
( ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>] ) implies ( (  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">y1</font>] ) &amp; ( ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>] ) implies <font color="Maroon" title="c1">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">y1</font> ) ) )
 
;<br/>
<b>hence </b><a NAME="E3:17_1_2_1"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">z</font>]
 ;<br/>


</div><b>end;</b></div><a NAME="E4:17_1_2"/><i><font color="Green" title="E14">A9</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">z1</font>, <font color="Olive" title="b3">z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z2</font>] holds <br/><font color="Olive" title="b2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z2</font>
 <b>by </b><i><a class="txt" href="partfun1.html#E2:17"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c2">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:17_1_2"/><i><font color="Green" title="E15">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E7:17_1_2"/><i><font color="Green" title="E16">A11</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="partfun1.html#E4:17_1_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="partfun1.html#E3:17_1_2"><i><font color="Green" title="E13">A8</font></i></a>);<br/></i><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b1">y</font>];<br/><b>consider </b><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:17_1_2"/><i><font color="Green" title="E17">A12</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i><b>set </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c2">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">X</font>;<br/><a NAME="E10:17_1_2"/><i><font color="Green" title="E18">A13</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="partfun1.html#E6:17_1_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/><a NAME="E11:17_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1_2_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_1_2_2"/><i><font color="Green" title="E19">A14</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 <b>and </b><br/><a NAME="E4:17_1_2_2"/><i><font color="Green" title="E20">A15</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:17_1_2_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/>
<a NAME="E6:17_1_2_2"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] )
 
<b>by </b><i><a class="txt" href="partfun1.html#E9:17_1_2"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="partfun1.html#E3:17_1_2_2"><i><font color="Green" title="E19">A14</font></i></a></i>;<br/>
<a NAME="E7:17_1_2_2"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>] )
 
<b>by </b><i><a class="txt" href="partfun1.html#E7:17_1_2"><i><font color="Green" title="E16">A11</font></i></a></i>;<br/>
<a NAME="E8:17_1_2_2"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">y</font>] )
 
<b>by </b><i><a class="txt" href="partfun1.html#E3:17_1_2_2"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="partfun1.html#E4:17_1_2_2"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E9:17_1_2_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="partfun1.html#E1:17"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c2">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c3">X</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="partfun1.html#E10:17_1_2"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="partfun1.html#E6"><i><font color="Green" title="E5">Lm3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font>;<br/><b>thus </b><a NAME="E13:17_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:17_1_2_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/>
<b>hence </b><a NAME="E2:17_1_2_3"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> implies ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] ) )
 <b>by </b><i><a class="txt" href="partfun1.html#E9:17_1_2"><i><font color="Green" title="E17">A12</font></i></a></i>;<br/>

<b>assume </b><a NAME="E3:17_1_2_3"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )
 ;<br/>

<a NAME="E4:17_1_2_3"/><b>then </b>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">g</font> )
 
<b>by </b><i><a class="txt" href="partfun1.html#E6:17_1_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="partfun1.html#E9:17_1_2"><i><font color="Green" title="E17">A12</font></i></a></i>;<br/>
<a NAME="E5:17_1_2_3"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1_2_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E14:17_1_2"/><i><font color="Green" title="E19">A16</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 ;<br/><a NAME="E15:17_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/><a NAME="E16:17_1_2"/><b>then </b>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )
 <b>by </b><i><a class="txt" href="partfun1.html#E9:17_1_2"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="partfun1.html#E14:17_1_2"><i><font color="Green" title="E19">A16</font></i></a></i>;<br/><a NAME="E17:17_1_2"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>]
 <b>by </b><i><a class="txt" href="partfun1.html#E7:17_1_2"><i><font color="Green" title="E16">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E18:17_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>]
 <b>by </b><i><a class="txt" href="partfun1.html#E14:17_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() st <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> iff ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>] ) ) ) &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font>] ) )
 <b>by </b><i><a class="txt" href="partfun1.html#E1:17_1"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>


</div>
