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<span class="kw">set </span><font color="Maroon" title="c1">A</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> 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">object</a>  ex <font color="Olive" title="b3">c</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">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; <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; (  for <font color="Olive" title="b4">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">c</font> iff ( <font color="Olive" title="b4">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="b4">f</font>] ) ) ) );<br/>
<div><span class="lab"><font color="Green" title="E35">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:73_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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_1"/>
<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>()
 <span class="kw">and </span><br/><a NAME="E4:73_1_1"/>
<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>()
 <span class="kw">and </span><br/><a NAME="E5:73_1_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,$1];<br/><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E7:73_1_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" onmouseover="rs('xboole_0/S1')" onmouseout="rh()">XBOOLE_0:sch 1</a>();<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">y</font>]</span><br/><span class="kw">thus </span><a NAME="E9:73_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E7:73_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c2">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:73_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.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>
 <span class="kw">and </span><br/><a NAME="E4:73_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" onmouseover="rs('classes1/S1')" onmouseout="rh()">CLASSES1:sch 1</a>(<a class="txt" href="#E1:73_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E37">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> for <font color="Olive" title="b3">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b1">a</font>,<font color="Olive" title="b2">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] &amp; <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI: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>)) ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() implies  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b1">f</font>] &amp; <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI: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>)) ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73_1_2"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E2:73_1_2"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b1">f</font>] &amp; <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI: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>)) ) )</span><br/><a NAME="E3:73_1_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_2"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E2:73_1_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/><a NAME="E4:73_1_2"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<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>,<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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a9</font>, <font color="Maroon" title="c6">b9</font>, <font color="Maroon" title="c7">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:73_1_2"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
( <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">a9</font>,<font color="Maroon" title="c6">b9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">a9</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">b9</font>)) )
 <span class="kw">and </span><br/><a NAME="E7:73_1_2"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
 for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">a9</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">b9</font>)) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>,<font color="Olive" title="b1">f</font>] ) )
 ;<br/><a NAME="E8:73_1_2"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/><a NAME="E9:73_1_2"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c8">f</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI: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="c8">f</font>] &amp; <font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI: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>)) ) )</span><br/><span class="kw">thus </span><a NAME="E10:73_1_2"/>
( <font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI: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="c8">f</font>] &amp; <font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI: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>)) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E8:73_1_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="txt" href="#E9:73_1_2"><span class="lab"><font color="Green" title="E43">A13</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">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><span class="lab"><font color="Green" title="E38">A14</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) &amp; <font color="Olive" title="b5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>) holds <br/><font color="Olive" title="b5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>)</span></a><div class="add"><span class="kw">let </span><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>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI: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>) holds <br/><font color="Olive" title="b2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI: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>)</span><br/><span class="kw">let </span><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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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>) implies <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="tarski.html#R2" title="TARSKI: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>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73_1_3"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI: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>)
 <span class="kw">and </span><br/><a NAME="E2:73_1_3"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI: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>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="tarski.html#R2" title="TARSKI: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>)</span><br/><a NAME="E3:73_1_3"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
<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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E1:73_1_3"><span class="lab"><font color="Green" title="E39">A15</font></span></a></span>;<br/><a NAME="E4:73_1_3"/><span class="lab"><font color="Green" title="E42">A18</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI: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>))
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E1:73_1_3"><span class="lab"><font color="Green" title="E39">A15</font></span></a></span>;<br/><a NAME="E5:73_1_3"/><span class="lab"><font color="Green" title="E43">A19</font></span>: 
<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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E2:73_1_3"><span class="lab"><font color="Green" title="E40">A16</font></span></a></span>;<br/><a NAME="E6:73_1_3"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI: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>))
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E2:73_1_3"><span class="lab"><font color="Green" title="E40">A16</font></span></a></span>;<br/><a NAME="E7:73_1_3"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_3"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="yellow18.html#T30" target="_self" onmouseover="rs('yellow18/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E8:73_1_3"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_3"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="yellow18.html#T30" target="_self" onmouseover="rs('yellow18/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E9:73_1_3"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="yellow18.html#T30" target="_self" onmouseover="rs('yellow18/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E10:73_1_3"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="yellow18.html#T30" target="_self" onmouseover="rs('yellow18/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E11:73_1_3"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T26" onmouseover="rs('relat_1/T26')" onmouseout="rh()">RELAT_1:26</a></span>;<br/><a NAME="E12:73_1_3"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73_1_3"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E8:73_1_3"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E9:73_1_3"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/><a NAME="E13:73_1_3"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_1_3"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E11:73_1_3"><span class="lab"><font color="Green" title="E49">A25</font></span></a></span>;<br/><a NAME="E14:73_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A27</font></span>: 
<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="tarski.html#R2" title="TARSKI: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>))
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:73_1_3"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/><a NAME="E15:73_1_3"/>
<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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:73_1_3"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E5:73_1_3"><span class="lab"><font color="Green" title="E43">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:73_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="tarski.html#R2" title="TARSKI: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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E14:73_1_3"><span class="lab"><font color="Green" title="E51">A27</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:73_1"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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="Olive" title="b1">a</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b2">b</font>))
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a></span>;<br/>
<a NAME="E8:73_1"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
 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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="tarski.html#R2" title="TARSKI: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>)</span><br/>

<a NAME="E1:73_1_4"/><span class="lab"><font color="Green" title="E41">A30</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>)
 
;<br/>
<a NAME="E2:73_1_4"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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>)
 
;<br/>
<a NAME="E3:73_1_4"/><span class="lab"><font color="Green" title="E43">A32</font></span>: 
<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>)]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E34">A2</font></span></a></span>;<br/>
<a NAME="E4:73_1_4"/>
 <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="tarski.html#R2" title="TARSKI: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>))
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_4"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E2:73_1_4"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:73_1_4"/>
 <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="tarski.html#R2" title="TARSKI: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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E3:73_1_4"><span class="lab"><font color="Green" title="E43">A32</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E10:73_1"/><span class="lab"><font color="Green" title="E41">A33</font></span>: 
 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>()
 <span class="kw">and </span><br/><a NAME="E11:73_1"/><span class="lab"><font color="Green" title="E42">A34</font></span>: 
 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>)
 <span class="kw">and </span><br/><a NAME="E12:73_1"/><span class="lab"><font color="Green" title="E43">A35</font></span>: 
 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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="yellow18.html#S16" target="_self" onmouseover="rs('yellow18/S16')" onmouseout="rh()">YELLOW18:sch 16</a>(<a class="txt" href="#E6:73_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E7:73_1"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="txt" href="#E8:73_1"><span class="lab"><font color="Green" title="E40">A29</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c3">C</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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>() &amp; (  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>) ) &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) iff ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI: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>] ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:73_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>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_1"><span class="lab"><font color="Green" title="E41">A33</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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>) ) &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) iff ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI: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>] ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E14:73_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73_1"><span class="lab"><font color="Green" title="E42">A34</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) iff ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI: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>] ) )</span><br/>

<span class="kw">let </span><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>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b1">f</font>] ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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>] ) )</span><br/>



<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_1"><span class="lab"><font color="Green" title="E41">A33</font></span></a></span>;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:73_1"><span class="lab"><font color="Green" title="E43">A35</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E17:73_1"/>
( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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>] ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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