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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">B</font> <a href="euclid_6.html#NR1" title="EUCLID_6:NR.1">is_a_triangle</a>  implies  <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">B</font> <a href="euclid_6.html#NR1" title="EUCLID_6:NR.1">is_a_triangle</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:21"/>
 <a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">i0</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:21"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_7.html#T7" onmouseover="rs('borsuk_7/T7')" onmouseout="rh()">BORSUK_7:7</a></span>;<br/>
<a NAME="E5:21"/>
( <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a>, <a class="ref" href="euclid11.html#T2" target="_self" onmouseover="rs('euclid11/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E6:21"/><span class="kw">then </span>
( <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a>  &amp; <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E7:21"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T5" onmouseover="rs('comptrig/T5')" onmouseout="rh()">COMPTRIG:5</a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/>
<a NAME="E8:21"/><span class="kw">then </span>
( <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> &amp; <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T5" onmouseover="rs('comptrig/T5')" onmouseout="rh()">COMPTRIG:5</a></span>;<br/>
<a NAME="E9:21"/><span class="kw">then </span>
( 3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> &amp; <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T88" onmouseover="rs('xcmplx_1/T88')" onmouseout="rh()">XCMPLX_1:88</a>, <a class="ref" href="comptrig.html#T5" onmouseover="rs('comptrig/T5')" onmouseout="rh()">COMPTRIG:5</a></span>;<br/>
<a NAME="E10:21"/><span class="kw">then </span>
( 3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>;<br/>
<a NAME="E11:21"/><span class="kw">then </span>
<font color="Maroon" title="c4">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="euclid11.html#E8"><span class="lab"><font color="Green" title="E7">Lm4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:21"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="euclid_6.html#T20" onmouseover="rs('euclid_6/T20')" onmouseout="rh()">EUCLID_6:20</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
