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<b>let </b><font color="Maroon">S</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Y</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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">Y</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>set </b><font color="Maroon">SS</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #);<br/>
<a NAME="E3:23"/><i><font color="Green">E48</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> #)</span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:23"/><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:23"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E7:23"/><i><font color="Green">E50</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E8:23"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">gg</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E10:23"/><i><font color="Green">E64</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> #)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:23"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #)</span>)</span>,<font color="Maroon">A</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E66</font></i>: <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #) <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #)</span>)</span>,<font color="Maroon">A</font></span><a href="borsuk_1.html#K6">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>
;<br/>
<a NAME="E13:23"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #),<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> #)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:23"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, , , , <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>


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