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<span class="kw">let </span><font color="Maroon" title="c4">S</font>, <font color="Maroon" title="c5">T</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">S</font><br/> for <font color="Olive" title="b3">t1</font>, <font color="Olive" title="b4">t2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font>  st <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">s1</font>,<font color="Olive" title="b3">t1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>,<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b2">s2</font>,<font color="Olive" title="b4">t2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/><font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">s1</font>, <font color="Maroon" title="c7">s2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font>  st <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Olive" title="b1">t1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>,<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Olive" title="b2">t2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c7">s2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c8">t1</font>, <font color="Maroon" title="c9">t2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>,<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  implies <font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c7">s2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  )</span><br/>











<span class="kw">given </span><font color="Maroon" title="c10">L</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> ,<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><a NAME="E2:15"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c10">L</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E3:15"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c8">t1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:15"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">t2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c7">s2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> </span><br/>

<span class="kw">take </span><font color="Maroon" title="c11">f</font> =  <a href="topalg_4.html#K4" title="TOPALG_4:func.4">pr1</a> <font color="Maroon" title="c10">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s1</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:15"/>
<font color="Maroon" title="c11">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="topalg_4.html#T9" target="_self" title="TOPALG_4:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s1</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> )</span><br/>

<a NAME="E6:15"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">f</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 <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></span>;<br/>
<a NAME="E7:15"/><span class="kw">then </span>
<font color="Maroon" title="c2">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">L</font>
 
;<br/>
<span class="kw">hence </span><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></span>
<br/>.= 
<font color="Maroon" title="c6">s1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font></span><br/>

<a NAME="E9:15"/>
<font color="Maroon" title="c3">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E17">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></span>
<br/>.= 
<font color="Maroon" title="c7">s2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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