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<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">s1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s2</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="E1:9"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1"/>
( <font color="Maroon">s1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s2</font> or <font color="Maroon">s2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s1</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><b>case </b></a><a NAME="E1:9_1_1_1"/>
<font color="Maroon">s1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s2</font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_1"/><b>then </b>
( <font color="Maroon">s1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s2</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s2</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s2</font> )
 <b>by </b><i><a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_1"/>
 ex <font color="Olive">rc</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">rc</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s2</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">rc</font> &amp; <font color="Olive">rc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2"><b>case </b></a><a NAME="E1:9_1_1_2"/>
<font color="Maroon">s2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s1</font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_2"/><b>then </b>
( <font color="Maroon">s2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s2</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s2</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s1</font> )
 <b>by </b><i><a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_2"/>
 ex <font color="Olive">rc</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">rc</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s2</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">rc</font> &amp; <font color="Olive">rc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
 ex <font color="Olive">r1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s2</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 )
 ;<br/>


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