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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">  for <font color="Olive" title="b1">T0</font> being   <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">T0</font>  ex <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Olive" title="b1">T0</font> st <font color="Olive" title="b2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">T0</font> be   <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</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">  for <font color="Olive" title="b1">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">T0</font>  ex <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c2">T0</font> st <font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">T0</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">  ex <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c2">T0</font> st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">F</font> =  <a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">R</font> =  <a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">TR</font> =  <a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</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">D1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">D1</font> );<br/>
<a NAME="E1:24"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 for <font color="Olive" title="b1">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">C</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="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">C</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1"/>
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</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 <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">C</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E3:24_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c8">p</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#T3" target="_self" onmouseover="rs('t_0topsp/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E4:24_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c7">C</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<a NAME="E6:24_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="t_0topsp.html#T12" target="_self" onmouseover="rs('t_0topsp/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:24_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">C</font>,<font color="Olive" title="b1">y</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E16">A3</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="E2:24"/>
 ex <font color="Olive" title="b1">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T0</font> st <br/> for <font color="Olive" title="b2">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C</font>,<font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">C</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T0</font><span class="kw"> such that </span><br/><a NAME="E4:24"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 for <font color="Olive" title="b1">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C</font>]
 ;<br/>
<a NAME="E5:24"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Olive" title="b1">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</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">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c8">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span><br/>

<a NAME="E1:24_2"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c8">p</font>) is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#T3" target="_self" onmouseover="rs('t_0topsp/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E2:24_2"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c8">p</font>),<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c8">p</font>)</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E15">A4</font></span></a></span>;<br/>
<a NAME="E3:24_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c8">p</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c8">p</font>)</span>)</span>
 
;<br/>
<a NAME="E4:24_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c8">p</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#T12" target="_self" onmouseover="rs('t_0topsp/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:24_2"/>
<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c8">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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">reconsider </span><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c7">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c2">T0</font> ;<br/>
<a NAME="E7:24"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E8:24"/>
 for <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T0</font>  st <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/><font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T0</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="c9">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_3"/>
<font color="Maroon" title="c9">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="tops_2.html#T43" onmouseover="rs('tops_2/T43')" onmouseout="rh()">TOPS_2:43</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">V</font> = <font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font>;<br/>
<a NAME="E3:24_3"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font></span>)</span> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font></span>)</span> iff <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font> )</span><br/>

<div><span class="kw">hereby </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"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:24_3_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font></span>)</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">C</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:24_3_1_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font>
 <span class="kw">and </span><br/><a NAME="E4:24_3_1_1"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c12">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E6:24_3_1_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c13">p</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_3_1_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="t_0topsp.html#T3" target="_self" onmouseover="rs('t_0topsp/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E7:24_3_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_3_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E6:24_3_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a></span>;<br/><a NAME="E8:24_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E9:24_3_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_3_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E6:24_3_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="eqrel_1.html#T19" onmouseover="rs('eqrel_1/T19')" onmouseout="rh()">EQREL_1:19</a></span>;<br/><a NAME="E10:24_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_3_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E6:24_3_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="eqrel_1.html#T35" onmouseover="rs('eqrel_1/T35')" onmouseout="rh()">EQREL_1:35</a></span>;<br/><a NAME="E11:24_3_1_1"/>
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_3_1_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/><a NAME="E12:24_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E3:24_3_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E10:24_3_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E13:24_3_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_3_1_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</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">assume </span><a NAME="E2:24_3_1"/><span class="lab"><font color="Green" title="E19">A13</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font></span>)</span></span><br/>

<a NAME="E3:24_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E4:24_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A14</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E2:24_3_1"><span class="lab"><font color="Green" title="E19">A13</font></span></a></span>;<br/>
<a NAME="E5:24_3_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>) is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_3_1"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="ref" href="t_0topsp.html#T3" target="_self" onmouseover="rs('t_0topsp/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E6:24_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A15</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_3_1"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="funct_2.html#T38" onmouseover="rs('funct_2/T38')" onmouseout="rh()">FUNCT_2:38</a></span>;<br/>
<a NAME="E7:24_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_3_1"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="ref" href="eqrel_1.html#T20" onmouseover="rs('eqrel_1/T20')" onmouseout="rh()">EQREL_1:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:24_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_3_1"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</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="E4:24_3"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>;<br/>
<a NAME="E5:24_3"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E18">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:24_3"/>
<font color="Maroon" title="c8">h</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c9">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#T2" target="_self" onmouseover="rs('t_0topsp/T2')" onmouseout="rh()">Th2</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c8">h</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c2">T0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="tops_2.html#T43" onmouseover="rs('tops_2/T43')" onmouseout="rh()">TOPS_2:43</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>
<a NAME="E10:24"/>
 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>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> implies <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_4"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span><br/>

<a NAME="E2:24_4"/><span class="kw">then </span>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" onmouseover="rs('eqrel_1/D3')" onmouseout="rh()">EQREL_1:def 3</a></span>;<br/>
<a NAME="E3:24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A17</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D7" onmouseover="rs('borsuk_1/D7')" onmouseout="rh()">BORSUK_1:def 7</a></span>;<br/>
<a NAME="E4:24_4"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_4"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="ref" href="t_0topsp.html#T4" target="_self" onmouseover="rs('t_0topsp/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:24_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/>
<a NAME="E6:24_4"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>),<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>)</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E3:24_4"><span class="lab"><font color="Green" title="E19">A17</font></span></a></span>;<br/>
<a NAME="E7:24_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="t_0topsp.html#K1" title="T_0TOPSP:func.1">Indiscernibility</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c11">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_4"><span class="lab"><font color="Green" title="E20">A18</font></span></a></span>;<br/>
<a NAME="E8:24_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_4"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="ref" href="t_0topsp.html#T12" target="_self" onmouseover="rs('t_0topsp/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:24_4"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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">hence </span><a NAME="E11:24"/>
 ex <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" title="T_0TOPSP:func.3">T_0-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c2">T0</font> st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" title="T_0TOPSP:func.4">T_0-canonical_map</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</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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