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<b>let </b><font color="Maroon">GX</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>  ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<font color="Maroon">GX</font> st <br/>( <font color="Olive">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 1 )
 ;<br/>

<a NAME="E2:3"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>  ex <font color="Olive">GY</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> st <br/>( <font color="Olive">GY</font> is <a href="connsp_1.html#V1">connected</a> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">GY</font>,<font color="Maroon">GX</font> st <br/>( <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<font color="Maroon">GX</font><b> such that </b><br/><a NAME="E2:3_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E3:3_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i/>;<br/><a NAME="E4:3_1_1"/>
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> )
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:3_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:3_1_1"/>
 ex <font color="Olive">GY</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> st <br/>( <font color="Olive">GY</font> is <a href="connsp_1.html#V1">connected</a> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">GY</font>,<font color="Maroon">GX</font> st <br/>( <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> ) )
 <b>by </b><i>, <a class="ref" href="treal_1.html#T22">TREAL_1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:3_1"/>
 ex <font color="Olive">GY</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> st <br/>( <font color="Olive">GY</font> is <a href="connsp_1.html#V1">connected</a> &amp;  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">GY</font>,<font color="Maroon">GX</font> st <br/>( <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
<font color="Maroon">GX</font> is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i/>;<br/>


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