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

<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</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>;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a>
 ;<br/>

<a NAME="E2:29"/><i><font color="Green" title="E16">A</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</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>  &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<b>let </b><font color="Maroon" title="c3">f</font> be   <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">S</font>;<br/>

<b>assume </b><a NAME="E3:29"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ;<br/>

<a NAME="E4:29"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
 <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:29"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> 
 ;<br/>
<a NAME="E7:29"/>
<font color="Maroon" title="c4">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:29"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="waybel18.html#E6:29"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E9:29"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c5">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E10:29"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:29"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="waybel18.html#D5" target="_self" title="WAYBEL18:def.5">Def5</a></i>;<br/>
<a NAME="E11:29"/>
<font color="Maroon" title="c4">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:29"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="waybel18.html#E6:29"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="tops_2.html#T70" title="TOPS_2:th.70">TOPS_2:70</a></i>;<br/>
<a NAME="E12:29"/><b>then </b><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">p</font> =  <a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E14:29"/>
<font color="Maroon" title="c6">p</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T98" title="TMAP_1:th.98">TMAP_1:98</a></i>;<br/>
<a NAME="E15:29"/><b>then </b><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:29"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<i><font color="Green" title="E24">A9</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<b>take </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">h</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel18.html#D8" target="_self" title="WAYBEL18:def.8">WAYBEL18:def 8</a><br/>

<a NAME="E17:29"/><i><font color="Green" title="E25">A10</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<a NAME="E18:29"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E19:29"/><b>then </b><i><font color="Green" title="E26">A11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:29"/><i><font color="Green" title="E27">A12</font></i>: 
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> holds  <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_2_1"/>
<font color="Maroon" title="c9">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#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font>
 ;<br/><a NAME="E2:29_2_1"/><b>then </b><i><font color="Green" title="E28">A13</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:29_2_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:29_2_1"/><b>then </b><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E2:29_2_1"><i><font color="Green" title="E28">A13</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c9">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E2:29_2_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c10">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E4:29_2_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="tmap_1.html#T95" title="TMAP_1:th.95">TMAP_1:95</a></i>
;<br/><b>hence </b><a NAME="E7:29_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E16:29"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="waybel18.html#E2:29_2_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:29_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font>
 ;<br/>

<a NAME="E3:29_2"/><b>then </b><i><font color="Green" title="E28">A15</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E4:29_2"/><b>then </b><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E16:29"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:29_2"/><b>then </b><i><font color="Green" title="E30">A17</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E4:29_2"><i><font color="Green" title="E29">A16</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c9">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E16:29"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="waybel18.html#E3:29_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c10">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E5:29_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="tmap_1.html#T95" title="TMAP_1:th.95">TMAP_1:95</a></i>
;<br/>
<b>hence </b><a NAME="E8:29_2"/>
<font color="Maroon" title="c9">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#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E16:29"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="waybel18.html#E3:29_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:29"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c7">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><a NAME="E1:29_3"/>
<font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:29_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:29"><i><font color="Green" title="E16">A</font></i></a>, <a class="txt" href="waybel18.html#E15:29"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<a NAME="E3:29_3"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E20:29"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/>
<a NAME="E4:29_3"/><b>then </b>
<font color="Maroon" title="c5">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:29"><i><font color="Green" title="E16">A</font></i></a>, <a class="txt" href="waybel18.html#E9:29"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E5:29_3"/>
<font color="Maroon" title="c7">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T181" title="RELAT_1:th.181">RELAT_1:181</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E22:29"/>
<font color="Maroon" title="c7">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:29"><i><font color="Green" title="E16">A</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>

<a NAME="E23:29"/><i><font color="Green" title="E28">A18</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E24:29"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <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> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_4"/><i><font color="Green" title="E29">A19</font></i>: 
<font color="Maroon" title="c8">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>
 ;<br/>

<a NAME="E2:29_4"/><b>then </b><i><font color="Green" title="E30">A20</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:29_4"/><i><font color="Green" title="E31">A21</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:29_4"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:29_4"/><i><font color="Green" title="E32">A22</font></i>: 
 <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:29"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E5:29_4"/><i><font color="Green" title="E33">A23</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E6:29_4"/><i><font color="Green" title="E34">A24</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:29_4"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c8">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E2:29_4"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E6:29"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="waybel18.html#E10:29"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="waybel18.html#E17:29"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="waybel18.html#E6:29_4"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E4:29_4"><i><font color="Green" title="E32">A22</font></i></a>, <a class="txt" href="waybel18.html#E5:29_4"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="waybel18.html#E3:29_4"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="waybel18.html#E4:29_4"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E25:29"/><b>then </b><i><font color="Green" title="E29">A25</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E23:29"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c7">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">h</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">h</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a>, <a class="txt" href="waybel18.html#E25:29"><i><font color="Green" title="E29">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/>

<a NAME="E27:29"/><i><font color="Green" title="E30">A26</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_6"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_6"/><i><font color="Green" title="E31">A27</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>and </b><br/><a NAME="E4:29_6"/><i><font color="Green" title="E32">A28</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:29_6"/>
<font color="Maroon" title="c9">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="c2">S</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:29_6"><i><font color="Green" title="E31">A27</font></i></a></i>;<br/>
<a NAME="E6:29_6"/><b>then </b><i><font color="Green" title="E33">A29</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:29_6"/><b>then </b><i><font color="Green" title="E34">A30</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">h</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:29_6"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:29_6"><i><font color="Green" title="E32">A28</font></i></a>, <a class="txt" href="waybel18.html#E6:29_6"><i><font color="Green" title="E33">A29</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E9:29_6"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E19:29"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="waybel18.html#E7:29_6"><i><font color="Green" title="E34">A30</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c8">H</font> = <font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>;<br/>
<a NAME="E28:29"/><i><font color="Green" title="E31">A31</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E29:29"/><i><font color="Green" title="E32">A32</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E30:29"/><i><font color="Green" title="E33">B</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span> <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="E31:29"/><i><font color="Green" title="E34">A33</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_7"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:29_7"/><i><font color="Green" title="E35">A34</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 ;<br/>

<a NAME="E3:29_7"/><i><font color="Green" title="E36">A35</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E29:29"><i><font color="Green" title="E32">A32</font></i></a>, <a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<a NAME="E4:29_7"/><i><font color="Green" title="E37">A36</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:29_7"><i><font color="Green" title="E35">A34</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:29_7"/><b>then </b><i><font color="Green" title="E38">A37</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:29_7"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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">S</font>
 
;<br/>
<a NAME="E7:29_7"/><b>then </b><i><font color="Green" title="E39">A38</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green" title="E40">A39</font></i>: <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c9">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E4:29_7"><i><font color="Green" title="E37">A36</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</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="c9">y</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a>, <a class="txt" href="waybel18.html#E25:29"><i><font color="Green" title="E29">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/>
<a NAME="E9:29_7"/><b>then </b><i><font color="Green" title="E41">A40</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:29_7"><i><font color="Green" title="E36">A35</font></i></a>, <a class="txt" href="waybel18.html#E7:29_7"><i><font color="Green" title="E39">A38</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">pp</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E5:29_7"><i><font color="Green" title="E38">A37</font></i></a></i>;<br/>
<a NAME="E11:29_7"/>
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c9">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E7:29_7"><i><font color="Green" title="E39">A38</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:29_7"/><b>then </b><i><font color="Green" title="E42">A41</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E8:29_7"><i><font color="Green" title="E40">A39</font></i></a>, <a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<a NAME="E13:29_7"/><b>then </b><i><font color="Green" title="E43">A42</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E28:29"><i><font color="Green" title="E31">A31</font></i></a>, <a class="txt" href="waybel18.html#E4:29_7"><i><font color="Green" title="E37">A36</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c9">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E4:29_7"><i><font color="Green" title="E37">A36</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E13:29_7"><i><font color="Green" title="E43">A42</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">pp</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E4:29_7"><i><font color="Green" title="E37">A36</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E12:29_7"><i><font color="Green" title="E42">A41</font></i></a>, <a class="ref" href="tmap_1.html#T95" title="TMAP_1:th.95">TMAP_1:95</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</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="c9">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E4:29_7"><i><font color="Green" title="E37">A36</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="txt" href="waybel18.html#E25:29"><i><font color="Green" title="E29">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E2:29_7"><i><font color="Green" title="E35">A34</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E15:29_7"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E9:29_7"><i><font color="Green" title="E41">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:29"/><i><font color="Green" title="E35">A43</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D1" title="WAYBEL_1:def.1">WAYBEL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:29_8"/><i><font color="Green" title="E36">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ;<br/>

<a NAME="E2:29_8"/><i><font color="Green" title="E37">A45</font></i>: 
<font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
;<br/>
<a NAME="E3:29_8"/><b>then </b><i><font color="Green" title="E38">A46</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<a NAME="E4:29_8"/><i><font color="Green" title="E39">A47</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:29_8"><i><font color="Green" title="E37">A45</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:29_8"/><i><font color="Green" title="E40">A48</font></i>: 
<font color="Maroon" title="c10">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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
;<br/>
<a NAME="E6:29_8"/><b>then </b><i><font color="Green" title="E41">A49</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<a NAME="E7:29_8"/><i><font color="Green" title="E42">A50</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E5:29_8"><i><font color="Green" title="E40">A48</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c10">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E3:29_8"><i><font color="Green" title="E38">A46</font></i></a>, <a class="txt" href="waybel18.html#E6:29_8"><i><font color="Green" title="E41">A49</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x'</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c14">y'</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> <b>by </b><i><a class="txt" href="waybel18.html#E27:29"><i><font color="Green" title="E30">A26</font></i></a>, <a class="txt" href="waybel18.html#E3:29_8"><i><font color="Green" title="E38">A46</font></i></a>, <a class="txt" href="waybel18.html#E6:29_8"><i><font color="Green" title="E41">A49</font></i></a>, <a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x'</font> = 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="txt" href="waybel18.html#E10:29"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T95" title="TMAP_1:th.95">TMAP_1:95</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font>
<b>by </b><i><a class="txt" href="waybel18.html#E1:29_8"><i><font color="Green" title="E36">A44</font></i></a>, <a class="txt" href="waybel18.html#E4:29_8"><i><font color="Green" title="E39">A47</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E7:29_8"><i><font color="Green" title="E42">A50</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T95" title="TMAP_1:th.95">TMAP_1:95</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y'</font>
<b>by </b><i><a class="txt" href="waybel18.html#E10:29"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E11:29_8"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E12:29"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="waybel_1.html#D1" title="WAYBEL_1:def.1">WAYBEL_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E33:29"/>
 <a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T98" title="TMAP_1:th.98">TMAP_1:98</a></i>;<br/>
<a NAME="E34:29"/><b>then </b><i><font color="Green" title="E36">A51</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E9:29"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E35:29"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>  st <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:29_9"/>
<font color="Maroon" title="c9">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:29_9"/><b>then </b>
<font color="Maroon" title="c9">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E4:29_9"/><i><font color="Green" title="E37">A52</font></i>: 
<font color="Maroon" title="c10">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E5:29_9"/><i><font color="Green" title="E38">A53</font></i>: 
<font color="Maroon" title="c9">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Q</font> = <font color="Maroon" title="c10">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E7:29_9"/><i><font color="Green" title="E39">A54</font></i>: 
<font color="Maroon" title="c11">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:29_9"><i><font color="Green" title="E37">A52</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E8:29_9"/>
<font color="Maroon" title="c6">p</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T98" title="TMAP_1:th.98">TMAP_1:98</a></i>;<br/>
<a NAME="E9:29_9"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:29"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E10:29_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:29"><i><font color="Green" title="E16">A</font></i></a>, <a class="txt" href="waybel18.html#E7:29_9"><i><font color="Green" title="E39">A54</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<a NAME="E11:29_9"/><b>then </b><i><font color="Green" title="E40">A55</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E20:29"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/>
<a NAME="E12:29_9"/><i><font color="Green" title="E41">A56</font></i>: 
<span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_9_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_9_1_1"/><i><font color="Green" title="E42">A57</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:29_9_1_1"/><i><font color="Green" title="E43">A58</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 <b>and </b><br/><a NAME="E5:29_9_1_1"/><i><font color="Green" title="E44">A59</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:29_9_1_1"/>
<font color="Maroon" title="c13">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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:29_9_1_1"><i><font color="Green" title="E42">A57</font></i></a></i>;<br/><a NAME="E7:29_9_1_1"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">xx</font> = <font color="Maroon" title="c13">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E9:29_9_1_1"/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:29_9_1_1"><i><font color="Green" title="E42">A57</font></i></a>, <a class="ref" href="tmap_1.html#T95" title="TMAP_1:th.95">TMAP_1:95</a></i>;<br/><b>hence </b><a NAME="E10:29_9_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E4:29_9_1_1"><i><font color="Green" title="E43">A58</font></i></a>, <a class="txt" href="waybel18.html#E5:29_9_1_1"><i><font color="Green" title="E44">A59</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:29_9_1"/><i><font color="Green" title="E42">A60</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 ;<br/>

<a NAME="E3:29_9_1"/><b>then </b><i><font color="Green" title="E43">A61</font></i>: 
<font color="Maroon" title="c12">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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
;<br/>
<a NAME="E4:29_9_1"/><b>then </b><i><font color="Green" title="E44">A62</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:29_9_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:29_9_1"><i><font color="Green" title="E43">A61</font></i></a>, <a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">yy</font> = <font color="Maroon" title="c12">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E7:29_9_1"/>
<font color="Maroon" title="c13">yy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:29_9_1"><i><font color="Green" title="E42">A60</font></i></a>, <a class="ref" href="tmap_1.html#T95" title="TMAP_1:th.95">TMAP_1:95</a></i>;<br/>
<b>hence </b><a NAME="E8:29_9_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:29_9_1"><i><font color="Green" title="E42">A60</font></i></a>, <a class="txt" href="waybel18.html#E4:29_9_1"><i><font color="Green" title="E44">A62</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:29_9"/><i><font color="Green" title="E42">A63</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_9_2_1"/><i><font color="Green" title="E43">A64</font></i>: 
<font color="Maroon" title="c12">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#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">Q</font>
 ;<br/><a NAME="E2:29_9_2_1"/><b>then </b><i><font color="Green" title="E44">A65</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:29_9_2_1"/><i><font color="Green" title="E45">A66</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:29_9_2_1"><i><font color="Green" title="E43">A64</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E4:29_9_2_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:29_9_2_1"><i><font color="Green" title="E44">A65</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:29_9_2_1"/><b>then </b><i><font color="Green" title="E46">A67</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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="c2">S</font>
 ;<br/><a NAME="E6:29_9_2_1"/><b>then </b><i><font color="Green" title="E47">A68</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:29_9_2_1"/><i><font color="Green" title="E48">A69</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E5:29_9_2_1"><i><font color="Green" title="E46">A67</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c12">x</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c12">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E6:29_9_2_1"><i><font color="Green" title="E47">A68</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</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="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E2:29_9_2_1"><i><font color="Green" title="E44">A65</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="txt" href="waybel18.html#E25:29"><i><font color="Green" title="E29">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="waybel18.html#E1:29_9_2_1"><i><font color="Green" title="E43">A64</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/><a NAME="E9:29_9_2_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E7:29_9_2_1"><i><font color="Green" title="E48">A69</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:29_9_2_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/><a NAME="E11:29_9_2_1"/><b>then </b><i><font color="Green" title="E49">A70</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E5:29_9"><i><font color="Green" title="E38">A53</font></i></a>, <a class="txt" href="waybel18.html#E3:29_9_2_1"><i><font color="Green" title="E45">A66</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c12">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</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="c12">x</font>
<b>by </b><i><a class="txt" href="waybel18.html#E2:29_9_2_1"><i><font color="Green" title="E44">A65</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="txt" href="waybel18.html#E25:29"><i><font color="Green" title="E29">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="waybel18.html#E1:29_9_2_1"><i><font color="Green" title="E43">A64</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/><b>hence </b><a NAME="E13:29_9_2_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E6:29_9_2_1"><i><font color="Green" title="E47">A68</font></i></a>, <a class="txt" href="waybel18.html#E11:29_9_2_1"><i><font color="Green" title="E49">A70</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:29_9_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:29_9_2"/><i><font color="Green" title="E43">A71</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h</font>
 <b>and </b><br/><a NAME="E5:29_9_2"/><i><font color="Green" title="E44">A72</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 <b>and </b><br/><a NAME="E6:29_9_2"/><i><font color="Green" title="E45">A73</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:29_9_2"/><i><font color="Green" title="E46">A74</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">h</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:29_9_2"><i><font color="Green" title="E43">A71</font></i></a>, <a class="txt" href="waybel18.html#E6:29_9_2"><i><font color="Green" title="E45">A73</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:29_9_2"/><i><font color="Green" title="E47">A75</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E5:29_9"><i><font color="Green" title="E38">A53</font></i></a>, <a class="txt" href="waybel18.html#E5:29_9_2"><i><font color="Green" title="E44">A72</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:29_9_2"/>
<font color="Maroon" title="c13">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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E5:29_9_2"><i><font color="Green" title="E44">A72</font></i></a></i>;<br/>
<a NAME="E10:29_9_2"/><b>then </b><i><font color="Green" title="E48">A76</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<a NAME="E11:29_9_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E19:29"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="waybel18.html#E7:29_9_2"><i><font color="Green" title="E46">A74</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c13">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> <b>by </b><i><a class="txt" href="waybel18.html#E27:29"><i><font color="Green" title="E30">A26</font></i></a>, <a class="txt" href="waybel18.html#E10:29_9_2"><i><font color="Green" title="E48">A76</font></i></a>, <a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font> = 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E10:29"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</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="c12">x</font>
<b>by </b><i><a class="txt" href="waybel18.html#E19:29"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="waybel18.html#E7:29_9_2"><i><font color="Green" title="E46">A74</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="txt" href="waybel18.html#E25:29"><i><font color="Green" title="E29">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E6:29_9_2"><i><font color="Green" title="E45">A73</font></i></a>, <a class="txt" href="waybel18.html#E7:29_9_2"><i><font color="Green" title="E46">A74</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">b</font>
<b>by </b><i><a class="txt" href="waybel18.html#E10:29"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E14:29_9_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E12:29"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="waybel18.html#E8:29_9_2"><i><font color="Green" title="E47">A75</font></i></a>, <a class="ref" href="waybel_1.html#D1" title="WAYBEL_1:def.1">WAYBEL_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:29_9_2"/>
<font color="Maroon" title="c12">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#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">Q</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E19:29"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="waybel18.html#E7:29_9_2"><i><font color="Green" title="E46">A74</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:29_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E12:29_9"><i><font color="Green" title="E41">A56</font></i></a>, <a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></i>;<br/>
<b>hence </b><a NAME="E15:29_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E31:29"><i><font color="Green" title="E34">A33</font></i></a>, <a class="txt" href="waybel18.html#E32:29"><i><font color="Green" title="E35">A43</font></i></a>, <a class="txt" href="waybel18.html#E11:29_9"><i><font color="Green" title="E40">A55</font></i></a>, <a class="txt" href="waybel18.html#E13:29_9"><i><font color="Green" title="E42">A63</font></i></a>, <a class="ref" href="tops_2.html#T67" title="TOPS_2:th.67">TOPS_2:67</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E30:29"><i><font color="Green" title="E33">B</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<a NAME="E37:29"/><b>then </b>
<font color="Maroon" title="c5">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4" title="TMAP_1:func.4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="waybel18.html#E29:29"><i><font color="Green" title="E32">A32</font></i></a>, <a class="txt" href="waybel18.html#E31:29"><i><font color="Green" title="E34">A33</font></i></a>, <a class="txt" href="waybel18.html#E32:29"><i><font color="Green" title="E35">A43</font></i></a>, <a class="txt" href="waybel18.html#E34:29"><i><font color="Green" title="E36">A51</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E38:29"/>
 <a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c1">T</font> <a href="t_0topsp.html#R1" title="T_0TOPSP:pred.1">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a></i>;<br/>


</div>
