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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Z</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font>;<br/>



<b>assume </b><a NAME="E1:30"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E3:30"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Z</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">Xf</font> =  <a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">Xg</font> =  <a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">XY</font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>
<b>set </b><font color="Maroon">XZ</font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Z</font>;<br/>
<a NAME="E4:30"/><i><font color="Green">E59</font></i>: 
(  <a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a> &amp;  <a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">a</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/><a NAME="E2:30_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K1">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font>
<b>by </b><i>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K1">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">h</font>
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:30"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Z</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">a</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Z</font> <b>by </b><i/>;<br/><a NAME="E2:30_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K1">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font>
<b>by </b><i>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K1">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">h</font>
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Z</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:30"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E9:30"/>
 <a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, , <a class="ref" href="yellow16.html#T17">YELLOW16:17</a></i>;<br/>


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