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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>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> , <font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> ;<br/>







<b>assume </b><a NAME="E1:21"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:21"/><i><font color="Green">E30</font></i>: 
( ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> ) &amp; <font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E5:21"/><i><font color="Green">E31</font></i>: 
( ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font> ) &amp; <font color="Maroon">g2</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E7:21"/><i><font color="Green">E35</font></i>: 
( ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> ) &amp; <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E8:21"/>
for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:21_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E3"><i><font color="Green">Lemma27</font></i></a></i>;<br/>
<a NAME="E2:21_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:21"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> ) &amp; <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3"><i><font color="Green">Lemma27</font></i></a></i>;<br/>


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