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<span class="kw">let </span><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a>  iff ( <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  ) )</span><br/>





<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  implies <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )</span><br/><a NAME="E2:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A2</font></span>: 
not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_6.html#D3" target="_self" title="EUCLID_6:def.3">Def3</a></span>;<br/><a NAME="E3:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T69" title="RLTOPSP1:th.69">RLTOPSP1:69</a></span>;<br/><a NAME="E4:41_1"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="euclid_6.html#D3" target="_self" title="EUCLID_6:def.3">Def3</a></span>;<br/><a NAME="E5:41_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T69" title="RLTOPSP1:th.69">RLTOPSP1:69</a></span>;<br/><span class="kw">hence </span><a NAME="E6:41_1"/>
<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )</span><br/><a NAME="E7:41_1"/>
not <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="euclid_6.html#D3" target="_self" title="EUCLID_6:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:41_1"/>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="euclid_6.html#T11" target="_self" title="EUCLID_6:th.11">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:41"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  or  not  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  or  not  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  or <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a>  )</span><br/>

<a NAME="E3:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:41"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  or  not  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  or <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a>  )</span><br/>

<a NAME="E5:41"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/>
<a NAME="E6:41"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:41"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  or <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a>  )</span><br/>

<a NAME="E8:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A10</font></span>: 
not <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="euclid_6.html#T8" target="_self" title="EUCLID_6:th.8">Th8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:41"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a> </span><br/>

<a NAME="E10:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A11</font></span>: 
not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="euclid_6.html#T8" target="_self" title="EUCLID_6:th.8">Th8</a></span>;<br/>
<a NAME="E11:41"/>
not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="euclid_6.html#T8" target="_self" title="EUCLID_6:th.8">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:41"/>
<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:41"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E10:41"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="euclid_6.html#D3" target="_self" title="EUCLID_6:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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