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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:76"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:76"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E3:76"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/>
<a NAME="E4:76"/><b>then </b>
( ( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font> or <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="absvalue.html#T1">ABSVALUE:1</a></i>;<br/>
<a NAME="E5:76"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T46">JGRAPH_1:46</a></i>;<br/>
<a NAME="E6:76"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="square_1.html#K3">^2</a>  &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a>, <a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/>
<a NAME="E7:76"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span></span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E8:76"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T66">EUCLID:66</a></i>;<br/>
<a NAME="E9:76"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/>
<a NAME="E10:76"/><b>then </b>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/>
<b>hence </b><a NAME="E11:76"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:76"><i><font color="Green">E55</font></i></a>, <a class="ref" href="topgen_5.html#D5" target="_self">Def5</a></i>;<br/>


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