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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">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Z</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">C</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:17"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">C</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Z</font> <b>by </b><i/>;<br/>
<a NAME="E4:17"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">Z</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">F</font> is <a href="struct_0.html#V3">empty</a>
 
;<br/>
<a NAME="E5:17"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">V</font> is <a href="struct_0.html#V3">empty</a>
 
;<br/>
<i><font color="Green">E43</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E7:17"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span> <b>by </b><i>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">Z</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">C</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">H</font> ;<br/>
<a NAME="E12:17"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">u</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T9">TOPMETR:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">H</font>,<font color="Maroon">G</font> ;<br/>
<a NAME="E14:17"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">j</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">k</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">G</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">G</font> ;<br/>
<a NAME="E16:17"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">w</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="jgraph_2.html#T2" target="_self">Th2</a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">C</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">u</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Z</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E19:17"/>
<font color="Maroon">v</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T4" target="_self">Th4</a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<b>hence </b><a NAME="E20:17"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Z</font> st <br/>( <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> )
 ;<br/>


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