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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><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="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1" target="_self">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:19_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K10" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E4:19_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K10" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K10" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="euclid.html#E1:19_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E6:19_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="euclid.html#K10" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="euclid.html#E6:19_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="euclid.html#K10" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T67">RVSUM_1:67</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:19"/>
 <a href="euclid.html#K10" target="_self">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="euclid.html#K10" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>


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