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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b2">X</font> being   <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b3">fs</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  st <font color="Olive" title="b2">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> holds <br/> <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Olive" title="b3">fs</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" 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="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b2">fs</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  st <font color="Olive" title="b1">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/> <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Olive" title="b2">fs</font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">fs</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  st <font color="Maroon" title="c3">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Olive" title="b1">fs</font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">fs</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c4">fs</font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )</span><br/>







<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> ($1,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c3">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> 
 <span class="kw">and </span><br/><a NAME="E2:29"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c4">fs</font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font></span><br/>

<a NAME="E3:29"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c3">X</font> is  <a href="zf_fund1.html#V4" title="ZF_FUND1:attr.4">closed_wrt_A4</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></span>;<br/>
<a NAME="E4:29"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c3">X</font> is  <a href="zf_fund1.html#V5" title="ZF_FUND1:attr.5">closed_wrt_A5</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="zf_fund1.html#D13" target="_self" title="ZF_FUND1:def.13">Def13</a></span>;<br/>
<a NAME="E5:29"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">fs</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">fs</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">fs</font> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">fs</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29_1"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">fs</font>
 <span class="kw">and </span><br/><a NAME="E2:29_1"/>
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">fs</font>
 <span class="kw">and </span><br/><a NAME="E3:29_1"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" 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="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_1_1"/>
( <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_1"/>
<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" 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="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:29_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a>, <a class="ref" href="zfmisc_1.html#T50" title="ZFMISC_1:th.50">ZFMISC_1:50</a></span>; <a class="txt" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_2"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" 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="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c7">r</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">A</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span> } </span> ;<br/><a NAME="E2:29_1_1_2"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
(  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E1:29_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="zf_fund1.html#T7" target="_self" title="ZF_FUND1:th.7">Th7</a></span>;<br/><a NAME="E3:29_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c5">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/><a NAME="E5:29_1_1_2"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span> } </span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:29_1_1_2_1"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span> } </span> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">q</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> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>) or <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1_2_1_1"/>
<font color="Maroon" title="c10">q</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> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:29_1_1_2_1_1"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">and </span><br/><a NAME="E4:29_1_1_2_1_1"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:29_1_1_2_1_1"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E6:29_1_1_2_1_1"/>
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_2_1_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E7:29_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</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> <font color="Maroon" title="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E5:29_1_1_2_1_1"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></span>;<br/>
<a NAME="E9:29_1_1_2_1_1"/>
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c12">s</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_2_1_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="grfunc_1.html#T7" title="GRFUNC_1:th.7">GRFUNC_1:7</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:29_1_1_2_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_2_1_1"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:29_1_1_2_1_1"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E7:29_1_1_2_1_1"><span class="lab"><font color="Green" title="E46">A14</font></span></a></span>; <a class="txt" 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">let </span><font color="Maroon" title="c10">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span> } </span>  or <font color="Maroon" title="c10">q</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> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:29_1_1_2_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">q</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> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E4:29_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p2"><a href="classes2.html#K6" title="CLASSES2:func.6">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="classes2.html#K6" title="CLASSES2:func.6">]</a></span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:29_1_1_2_1"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E6:29_1_1_2_1"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c10">q</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_2_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="grfunc_1.html#T5" title="GRFUNC_1:th.5">GRFUNC_1:5</a></span>;<br/>
<a NAME="E8:29_1_1_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_2_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="relat_1.html#T9" title="RELAT_1:th.9">RELAT_1:9</a></span>;<br/>
<a NAME="E9:29_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A18</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_1_1_2_1"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E10:29_1_1_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_1_2_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E11:29_1_1_2_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_2_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="relat_1.html#T9" title="RELAT_1:th.9">RELAT_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:29_1_1_2_1"/>
<font color="Maroon" title="c10">q</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> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_1_2_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>; <a class="txt" 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">reconsider </span><font color="Maroon" title="c10">x</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c2">a</font>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a></span>;<br/><a NAME="E7:29_1_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E2:29_1_1_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></span>;<br/><a NAME="E8:29_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A19</font></span>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E5:29_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="zf_fund1.html#D9" target="_self" title="ZF_FUND1:def.9">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">a</font>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c12">Z</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y9</font></span>)</span> where <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span> } </span> ;<br/><a NAME="E10:29_1_1_2"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y9</font></span>)</span> where <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span> } </span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:29_1_1_2_2"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y9</font></span>)</span> where <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span> } </span> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y9</font></span>)</span> where <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">q</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> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>) or <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y9</font></span>)</span> where <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1_2_2_1"/>
<font color="Maroon" title="c13">q</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> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y9</font></span>)</span> where <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:29_1_1_2_2_1"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">and </span><br/><a NAME="E4:29_1_1_2_2_1"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:29_1_1_2_2_1"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">A</font> = <font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font>;<br/>
<a NAME="E6:29_1_1_2_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</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="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T70" title="RELAT_1:th.70">RELAT_1:70</a></span>;<br/>
<a NAME="E7:29_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A23</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">C</font> = <font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E8:29_1_1_2_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T70" title="RELAT_1:th.70">RELAT_1:70</a></span>;<br/>
<a NAME="E9:29_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A24</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></span>
;<br/>
<a NAME="E11:29_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A25</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">y9</font> = <font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_1_1_2"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></span>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c6">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></span>
;<br/>
<a NAME="E14:29_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A26</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">x9</font> = <font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="zf_fund1.html#T1" target="_self" title="ZF_FUND1:th.1">Th1</a></span>;<br/>
<font color="Maroon" title="c14">g</font> = 
<font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T78" title="RELAT_1:th.78">RELAT_1:78</a></span>
;<br/>
<a NAME="E17:29_1_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c17">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:29_1_1_2_2_1"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y9</font></span>)</span> where <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E11:29_1_1_2_2_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a></span>; <a class="txt" 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">let </span><font color="Maroon" title="c13">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y9</font></span>)</span> where <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span> } </span>  or <font color="Maroon" title="c13">q</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> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:29_1_1_2_2"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Olive" title="b2">y9</font></span>)</span> where <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">q</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> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x9</font>, <font color="Maroon" title="c15">y9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E4:29_1_1_2_2"/><span class="lab"><font color="Green" title="E44">A27</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x9</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c15">y9</font>
 <span class="kw">and </span><br/><a NAME="E5:29_1_1_2_2"/><span class="lab"><font color="Green" title="E45">A28</font></span>: 
<font color="Maroon" title="c14">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">and </span><br/><a NAME="E6:29_1_1_2_2"/><span class="lab"><font color="Green" title="E46">A29</font></span>: 
<font color="Maroon" title="c15">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">e</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:29_1_1_2_2"/><span class="lab"><font color="Green" title="E47">A30</font></span>: 
<font color="Maroon" title="c14">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">e</font>
 <span class="kw">and </span><br/><a NAME="E9:29_1_1_2_2"/><span class="lab"><font color="Green" title="E48">A31</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 <span class="kw">and </span><br/><a NAME="E10:29_1_1_2_2"/><span class="lab"><font color="Green" title="E49">A32</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_1_2_2"><span class="lab"><font color="Green" title="E45">A28</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E12:29_1_1_2_2"/><span class="lab"><font color="Green" title="E50">A33</font></span>: 
<font color="Maroon" title="c15">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">and </span><br/><a NAME="E13:29_1_1_2_2"/><span class="lab"><font color="Green" title="E51">A34</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E14:29_1_1_2_2"/><span class="lab"><font color="Green" title="E52">A35</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_1_1_2_2"><span class="lab"><font color="Green" title="E46">A29</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c16">e</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">g</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E9:29_1_1_2_2"><span class="lab"><font color="Green" title="E48">A31</font></span></a>, <a class="txt" href="#E13:29_1_1_2_2"><span class="lab"><font color="Green" title="E51">A34</font></span></a>, <a class="ref" href="grfunc_1.html#T13" title="GRFUNC_1:th.13">GRFUNC_1:13</a></span>;<br/>
<a NAME="E16:29_1_1_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">e</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T12" title="RELAT_1:th.12">RELAT_1:12</a></span>;<br/>
<a NAME="E17:29_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A36</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="classes2.html#K7" title="CLASSES2:func.7">\/</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_1_1_2_2"><span class="lab"><font color="Green" title="E49">A32</font></span></a>, <a class="txt" href="#E14:29_1_1_2_2"><span class="lab"><font color="Green" title="E52">A35</font></span></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></span>;<br/>
<a NAME="E18:29_1_1_2_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_1_2_2"><span class="lab"><font color="Green" title="E48">A31</font></span></a>, <a class="txt" href="#E13:29_1_1_2_2"><span class="lab"><font color="Green" title="E51">A34</font></span></a>, <a class="ref" href="relat_1.html#T1" title="RELAT_1:th.1">RELAT_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:29_1_1_2_2"/>
<font color="Maroon" title="c13">q</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> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_2_2"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="txt" href="#E8:29_1_1_2_2"><span class="lab"><font color="Green" title="E47">A30</font></span></a>, <a class="txt" href="#E12:29_1_1_2_2"><span class="lab"><font color="Green" title="E50">A33</font></span></a>, <a class="txt" href="#E17:29_1_1_2_2"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>; <a class="txt" 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">hence </span><a NAME="E11:29_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E8:29_1_1_2"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="zf_fund1.html#D10" target="_self" title="ZF_FUND1:def.10">Def10</a></span>; <a class="txt" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E6:29"/>
(  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="zf_fund1.html#T3" target="_self" title="ZF_FUND1:th.3">Th3</a>, <a class="ref" href="funct_5.html#T57" title="FUNCT_5:th.57">FUNCT_5:57</a></span>;<br/>
<a NAME="E7:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A37</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></span>;<br/>
<a NAME="E8:29"/><span class="lab"><font color="Green" title="E39">A38</font></span>: 
<font color="Maroon" title="c4">fs</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<span class="kw">thus </span><a NAME="E9:29"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">fs</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="#E8:29"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E37">A5</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
