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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">v</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">q</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 3</span>)</span></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3 ) )</span><br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">v</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">q</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 3</span>)</span></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3 )</span><br/>

<b>let </b><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">v</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">q</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 3</span>)</span></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3 ) )</span><br/>



<b>assume </b><a NAME="E2:31"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">v</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">q</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 3</span>)</span></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3 )</span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span>
<b>by </b><i><a class="ref" href="polyeq_1.html#T14" target="_self" title="POLYEQ_1:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span>

;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green" title="E9">A3</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="txt" href="polyeq_1.html#E1:31"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="polyeq_1.html#E2:31"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green" title="E10">A4</font></i>: 
3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="polyeq_1.html#E2:31"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="polyeq_1.html#E2:31"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="polyeq_1.html#E4:31"><i><font color="Green" title="E9">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 3</span>)</span></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3</span><br/>

<b>thus </b><a NAME="E7:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 3</span>)</span></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3
 <b>by </b><i><a class="txt" href="polyeq_1.html#E5:31"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="newton.html#T12" title="NEWTON:th.12">NEWTON:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
