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<span class="kw">let </span><font color="Maroon" title="c1">s</font>, <font color="Maroon" title="c2">t</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t</font> implies for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">t</font> holds <br/><font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Olive" title="b1">r</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">t</font> holds <br/><font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Olive" title="b1">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">t</font> implies <font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Maroon" title="c3">r</font> )</span><br/>

<a NAME="E2:39"/>
<font color="Maroon" title="c2">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">s</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>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E3:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:39"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Maroon" title="c3">r</font></span><br/>

<a NAME="E5:39"/><span class="kw">then </span>
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">r</font>
 
;<br/>
<a NAME="E6:39"/><span class="kw">then </span>
<font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="newton.html#K5" title="NEWTON:func.5">!</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="newton.html#D3" target="_self" title="NEWTON:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:39"/>
<font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> <a href="newton.html#K7" title="NEWTON:func.7">choose</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="newton.html#D3" target="_self" title="NEWTON:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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