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<div class="add">

<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">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="E1:31"/><i><font color="Green">E27</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>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E3:31"/><i><font color="Green">E28</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">r2</font> ) &amp; <font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T5" target="_self">Th5</a></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>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E5:31"/><i><font color="Green">E42</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">r2</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">g3</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E7:31"/><i><font color="Green">E43</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font> ) &amp; <font color="Maroon">g3</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, <a class="txt" href="jgraph_2.html#E1:31"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">g4</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E9:31"/><i><font color="Green">E44</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font> ) &amp; <font color="Maroon">g4</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, <a class="txt" href="jgraph_2.html#E1:31"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">g5</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E11:31"/><i><font color="Green">E45</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">g5</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">r2</font> ) &amp; <font color="Maroon">g5</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>consider </b><font color="Maroon">g6</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E13:31"/><i><font color="Green">E46</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">g5</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">g6</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font> ) &amp; <font color="Maroon">g6</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, <a class="txt" href="jgraph_2.html#E3:31"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E14:31"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">g6</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">r1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">r2</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> )
 ;<br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:31_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:31_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">g3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r2</font></span>)</span> <a href="square_1.html#K3">^2</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:31_1"/>
<font color="Maroon">g4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2</font></span>)</span> <a href="square_1.html#K3">^2</a> 
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E6:31_1"/><b>then </b>
<font color="Maroon">g5</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r2</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><font color="Maroon">g6</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r2</font>

;<br/>
<b>hence </b><a NAME="E8:31_1"/>
<font color="Maroon">g6</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:31"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/><font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r2</font> ) &amp; <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i/>;<br/>


</div>
