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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K5">"</a>  is <a href="xreal_0.html#V1" target="_self">real</a>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_2_1_1_2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:6_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="xreal_0.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:6_2_1_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E5:6_2_1_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K5">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E6:6_2_1_1_2"/><i><font color="Green">E7</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5">XCMPLX_0:def 5</a></i>;<br/><a NAME="E7:6_2_1_1_2"/><i><font color="Green">E8</font></i>: 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="xreal_0.html#E6:6_2_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E8:6_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="xreal_0.html#E4:6_2_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E9:6_2_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/><a NAME="E10:6_2_1_1_2"/><b>then </b>
0 <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E7:6_2_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/><a NAME="E11:6_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="arytm_0.html#T23">ARYTM_0:23</a></i>;<br/><a NAME="E12:6_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K5">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="xreal_0.html#E1:6_2_1_1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="xreal_0.html#E4:6_2_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="xreal_0.html#E5:6_2_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="xreal_0.html#E2"><i><font color="Green">Lemma2</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><b>hence </b><a NAME="E13:6_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K5">"</a>  is <a href="xreal_0.html#V1" target="_self">real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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