<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c1">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c2">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">f1'</font> = <font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c5">f2'</font> = <font color="Maroon" title="c2">f2</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:46"/>
( <font color="Maroon" title="c4">f1'</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c5">f2'</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:46"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E5:46"/><i><font color="Green" title="E38">A2</font></i>: 
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c4">f1'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Maroon" title="c5">f2'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r2</font> holds <br/><font color="Maroon" title="c6">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r2</font> ) &amp; <font color="Maroon" title="c6">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T29" title="JGRAPH_2:th.29">JGRAPH_2:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:46"/><i><font color="Green" title="E39">A3</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <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 <span class="p1">(<span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E8:46"/><i><font color="Green" title="E40">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green" title="E41">A5</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="txt" href="borsuk_6.html#E3:46_1"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<a NAME="E5:46_1"/>
<font color="Maroon" title="c6">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:46"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<a NAME="E6:46_1"/><b>then </b>
<font color="Maroon" title="c6">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:46"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:46_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E3:46_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:46"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</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 <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c6">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<span class="p1">(<span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">A</font></span>)</span> <b>by </b><i><a class="txt" href="borsuk_6.html#E7:46"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="borsuk_6.html#E8:46"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E11:46"/>
<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D16" title="BORSUK_1:def.16">BORSUK_1:def 16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_6.html#E5:46"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="jgraph_1.html#T63" title="JGRAPH_1:th.63">JGRAPH_1:63</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">g</font>
;<br/>

<b>thus </b><a NAME="E13:46"/>
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Maroon" title="c2">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r2</font> holds <br/><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r2</font> ) &amp; <font color="Maroon" title="c9">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E5:46"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>


</div>
