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<b>let </b><font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive" title="b2">A</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <font color="Olive" title="b1">P</font> holds <br/><span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font>,<span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">P</font></span><br/><b>let </b><font color="Maroon" title="c7">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <font color="Maroon" title="c7">P</font> holds <br/><span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font>,<span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font></span><br/>





<b>set </b><font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c7">P</font>;<br/>
<b>let </b><font color="Maroon" title="c9">A</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <font color="Maroon" title="c7">P</font> implies <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font>,<span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font> )</span><br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <font color="Maroon" title="c7">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font>,<span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font></span><br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> holds <br/> <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> ) )
 
<b>by </b><i><a class="ref" href="jordan24.html#D1" target="_self" title="JORDAN24:def.1">Def1</a></i>;<br/>
<a NAME="E3:12"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:12"/>
( <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font> )
 <b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan24.html#D1" target="_self" title="JORDAN24:def.1">JORDAN24:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font> holds <br/> <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><br/>

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font> implies  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> )</span><br/>

<b>assume </b><a NAME="E5:12"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><br/>

<b>consider </b><font color="Maroon" title="c12">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:12"/><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> &amp; <font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">xx</font> = <font color="Maroon" title="c12">xx</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E7:12"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">yy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:12"/><i><font color="Green" title="E14">A6</font></i>: 
( <font color="Maroon" title="c14">yy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> &amp; <font color="Maroon" title="c14">yy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">yy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> )
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">yy</font> = <font color="Maroon" title="c14">yy</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E10:12"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">f</font> =  <a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span></span>)</span> ;<br/>
<a NAME="E13:12"/>
<font color="Maroon" title="c16">f</font> is <a href="jordan24.html#V1" title="JORDAN24:attr.1">isometric</a>
 
<b>by </b><i><a class="ref" href="jordan24.html#T2" target="_self" title="JORDAN24:th.2">Th2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">g</font> being  <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span><b> such that </b><br/><a NAME="E15:12"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c16">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c17">g</font>
 <b>by </b><i><a class="ref" href="jordan24.html#D2" target="_self" title="JORDAN24:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">p1'</font> = <font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c19">p2'</font> = <font color="Maroon" title="c6">p2</font>, <font color="Maroon" title="c20">xx'</font> = <font color="Maroon" title="c13">xx</font>, <font color="Maroon" title="c21">yy'</font> = <font color="Maroon" title="c15">yy</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/>
<a NAME="E17:12"/>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c13">xx</font>,<font color="Maroon" title="c15">yy</font>
 
<b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E7:12"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E10:12"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="jordan24.html#D1" target="_self" title="JORDAN24:def.1">Def1</a></i>;<br/>
<a NAME="E18:12"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">p1'</font>,<font color="Maroon" title="c19">p2'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c13">xx</font>,<font color="Maroon" title="c15">yy</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E19:12"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">p1'</font>,<font color="Maroon" title="c19">p2'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">xx'</font>,<font color="Maroon" title="c21">yy'</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E20:12"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p1'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">p2'</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">xx'</font>,<font color="Maroon" title="c21">yy'</font>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D3" title="VECTMETR:def.3">VECTMETR:def 3</a></i>;<br/>
<a NAME="E21:12"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p1'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">p2'</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">xx'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">yy'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D3" title="VECTMETR:def.3">VECTMETR:def 3</a></i>;<br/>
<a NAME="E22:12"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">xx'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">yy'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/>
<b>hence </b><a NAME="E23:12"/>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan24.html#K1" title="JORDAN24:func.1">Rotate</a> <font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="#E7:12"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E10:12"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E15:12"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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