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<span class="kw">let </span><font color="Maroon" title="c1">rf</font> be   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <font color="Maroon" title="c1">rf</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">F</font> = <font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">S</font> =  <a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span>;<br/>
<a NAME="E1:10"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun3.html#D5" title="PARTFUN3:def.5">PARTFUN3:def 5</a></span>;<br/>
<a NAME="E2:10"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <font color="Maroon" title="c1">rf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">rf</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">rf</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T11" title="VALUED_1:th.11">VALUED_1:11</a>, <a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <font color="Maroon" title="c1">rf</font></span>)</span> implies <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <font color="Maroon" title="c1">rf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">fx</font> = <font color="Maroon" title="c1">rf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E2:10_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
( <font color="Maroon" title="c5">fx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">fx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><span class="kw">assume </span><a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <font color="Maroon" title="c1">rf</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <font color="Maroon" title="c1">rf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><a NAME="E4:10_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fx</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp; <span class="p1">(<span class="default"><a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="partfun3.html#D5" title="PARTFUN3:def.5">PARTFUN3:def 5</a>, <a class="ref" href="valued_1.html#T11" title="VALUED_1:th.11">VALUED_1:11</a></span>;<br/><a NAME="E5:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A5</font></span>: 
( ( <span class="p1">(<span class="default"><a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fx</font> &amp; <font color="Maroon" title="c5">fx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <span class="p1">(<span class="default"><a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c5">fx</font> &amp; <font color="Maroon" title="c5">fx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="square_1.html#T22" title="SQUARE_1:th.22">SQUARE_1:22</a>, <a class="ref" href="square_1.html#T23" title="SQUARE_1:th.23">SQUARE_1:23</a></span>;<br/><a NAME="E6:10_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <font color="Maroon" title="c1">rf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">fx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/><span class="kw">hence </span><a NAME="E7:10_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <font color="Maroon" title="c1">rf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:10"/>
 <a href="partfun3.html#K1" title="PARTFUN3:func.1">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">rf</font> <a href="valued_1.html#K39" title="VALUED_1:func.39">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <font color="Maroon" title="c1">rf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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