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<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> 2</span>)</span>;<br/><b>let </b><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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>set </b><font color="Maroon">C</font> = <font color="Maroon">P</font>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E49</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:10"/>
<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="jordan24.html#R1" target="_self">realize-max-dist-in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E3:10"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> <a href="topreal6.html#K1">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="topreal6.html#K1">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E4:10"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:10"/>
( <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan24.html#D1" target="_self">JORDAN24:def 1</a><br/>

<b>let </b><font color="Maroon">x</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> 2</span>)</span>, <font color="Maroon">y</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> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E6:10"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> )
 ;<br/>

<b>consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:10"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">xx</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">yy</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:10"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">yy</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">yy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">yy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">yy</font> = <font color="Maroon">yy</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f</font> =  <a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span></span>)</span> ;<br/>
<a NAME="E14:10"/>
<font color="Maroon">f</font> is <a href="jordan24.html#V1" target="_self">isometric</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being  <a href="vectmetr.html#V3">isometric</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span><b> such that </b><br/><a NAME="E16:10"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p1'</font> = <font color="Maroon">p1</font>, <font color="Maroon">p2'</font> = <font color="Maroon">p2</font>, <font color="Maroon">xx'</font> = <font color="Maroon">xx</font>, <font color="Maroon">yy'</font> = <font color="Maroon">yy</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E18:10"/>
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="topreal6.html#K1">dist</a> <font color="Maroon">xx</font>,<font color="Maroon">yy</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E19:10"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="topreal6.html#K1">dist</a> <font color="Maroon">xx</font>,<font color="Maroon">yy</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E20:10"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">xx'</font>,<font color="Maroon">yy'</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E21:10"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">xx'</font>,<font color="Maroon">yy'</font>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D3">VECTMETR:def 3</a></i>;<br/>
<a NAME="E22:10"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">yy'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D3">VECTMETR:def 3</a></i>;<br/>
<a NAME="E23:10"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">yy'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>
<b>hence </b><a NAME="E24:10"/>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" target="_self">Rotate</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>


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