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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:69_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K6">/</a> 4 implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 ) )
 <b>and </b><br/><a NAME="E2:69_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K6">/</a> 4 implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 ) )
 ;<br/>

<a NAME="E3:69_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 ) or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K6">/</a> 4 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:69_1_2"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:69_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="borsuk_6.html#E1:69_1_2_1_1_1"><i><font color="Green">E54</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:69_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="real_1.html#K6">/</a> 4 )
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:69_1_2"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:69_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="borsuk_6.html#E1:69_1_2_1_1_2"><i><font color="Green">E54</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:69_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="real_1.html#K6">/</a> 4
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:69_1_2"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:69_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="borsuk_6.html#E1:69_1_2_1_1_3"><i><font color="Green">E54</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:69_1_2_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:69_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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