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<b>set </b><font color="Maroon" title="c1">A</font> = <font color="Maroon">F<sub>1</sub></font>();<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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( for <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 iff ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b2">b</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] ) ) ) );<br/>
<div><i><font color="Green" title="E39">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>now </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="E1:76_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:76_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><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><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,$1];<br/><b>consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:76_1_1"/><i><font color="Green" title="E41">A5</font></i>: 
for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">y</font> iff ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">f</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>take </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font>;<br/><b>thus </b><a NAME="E6:76_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">y</font>]
 <b>by </b><i><a class="txt" href="yellow18.html#E3:76_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="yellow18.html#E5:76_1_1"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:76_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E4:76_1"/><i><font color="Green" title="E40">A6</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">F</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="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="yellow18.html#E1:76_1"><i><font color="Green" title="E39">A3</font></i></a>);<br/></i>
<div><i><font color="Green" title="E41">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:76_1_2"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/><a NAME="E2:76_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">a'</font>, <font color="Maroon" title="c6">b'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:76_1_2"/><i><font color="Green" title="E42">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E5:76_1_2"/><i><font color="Green" title="E43">A9</font></i>: 
for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> iff ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">a'</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">b'</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>,<font color="Olive" title="b1">f</font>] ) )
 <b>by </b><i><a class="txt" href="yellow18.html#E4:76_1"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/><a NAME="E6:76_1_2"/><i><font color="Green" title="E44">A10</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font> )
 <b>by </b><i><a class="txt" href="yellow18.html#E4:76_1_2"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>let </b><font color="Maroon" title="c7">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E7:76_1_2"/>
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> iff ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">f</font>] &amp; <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>) ) )
 <b>by </b><i><a class="txt" href="yellow18.html#E5:76_1_2"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="yellow18.html#E6:76_1_2"><i><font color="Green" title="E44">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<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>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<div><i><font color="Green" title="E42">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>assume </b><a NAME="E1:76_1_3"/>
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>) )
 ;<br/><a NAME="E2:76_1_3"/><b>then </b><i><font color="Green" title="E43">A12</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">f</font>] &amp; <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">g</font>] &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">c</font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E5:76_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/><a NAME="E3:76_1_3"/><b>then </b>
(  <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) &amp;  <a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>) &amp;  <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>) &amp;  <a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <font color="Maroon" title="c7">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">c</font>) )
 <b>by </b><i><a class="ref" href="yellow18.html#T31" target="_self" title="YELLOW18:th.31">Th31</a></i>;<br/><a NAME="E4:76_1_3"/><b>then </b>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>) &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">c</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> )
 <b>by </b><i><a class="ref" href="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/><a NAME="E5:76_1_3"/><b>then </b>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">c</font>) )
 <b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:76_1_3"/><b>then </b>
( <font color="Maroon" title="c7">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">c</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">f</font>] )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:76"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="yellow18.html#E2:76_1_3"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><b>hence </b><a NAME="E7:76_1_3"/>
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E5:76_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E43">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>thus </b><a NAME="E1:76_1_4"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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>thus </b><a NAME="E1:76_1_4_1"/>
(  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">b</font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E5:76_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:76_1"/><i><font color="Green" title="E44">A14</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E1:76_1_5"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) )
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<a NAME="E2:76_1_5"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>, <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>)] &amp;  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) )
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:76"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:76_1_5"/>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E5:76_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">C</font> being  <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="altcat_1.html#V9" title="ALTCAT_1:attr.9">semi-functional</a> <a href="yellow18.html#V2" title="YELLOW18:attr.2">para-functional</a> <a href="yellow18.html#V3" title="YELLOW18:attr.3">set-id-inheriting</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a><b> such that </b><br/><a NAME="E10:76_1"/><i><font color="Green" title="E45">A15</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</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="E11:76_1"/><i><font color="Green" title="E46">A16</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font> holds   <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>)
 <b>and </b><br/><a NAME="E12:76_1"/><i><font color="Green" title="E47">A17</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)
 <b>from </b><i><a class="ref" href="yellow18.html#S16" target="_self" title="YELLOW18:sch.16">YELLOW18:sch 16</a>(<a class="txt" href="yellow18.html#E6:76_1"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="yellow18.html#E7:76_1"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="yellow18.html#E8:76_1"><i><font color="Green" title="E44">A14</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">C</font>
;<br/>

<b>thus </b><a NAME="E13:76_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="yellow18.html#E10:76_1"><i><font color="Green" title="E45">A15</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:76_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font> holds   <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E11:76_1"><i><font color="Green" title="E46">A16</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font> <b>by </b><i><a class="txt" href="yellow18.html#E10:76_1"><i><font color="Green" title="E45">A15</font></i></a></i>;<br/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> = 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>

<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E12:76_1"><i><font color="Green" title="E47">A17</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E17:76_1"/>
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> iff ( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">b</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">f</font>] ) )
 <b>by </b><i><a class="txt" href="yellow18.html#E5:76_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>


</div>
