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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">a9</font>, <font color="Maroon" title="c6">b9</font>, <font color="Maroon" title="c7">c9</font>, <font color="Maroon" title="c8">d9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</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>  holds   <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c7">c9</font>,<font color="Maroon" title="c8">d9</font>,<font color="Olive" title="b1">x</font> ) implies ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b9</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d9</font> ) )</span><br/>















<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3 = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="polyeq_1.html#E17"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a>, <a class="ref" href="power.html#T2" title="POWER:th.2">POWER:2</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>
;<br/>
<a NAME="E2:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A1</font></span>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></span>;<br/>
<span class="lab"><font color="Green" title="E6">A2</font></span>: 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1 = 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>
;<br/>
<span class="lab"><font color="Green" title="E7">A3</font></span>: 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 3 = 
2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>

;<br/>
<span class="kw">assume </span><a NAME="E5:23"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
for <font color="Olive" title="b1">x</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>  holds   <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c7">c9</font>,<font color="Maroon" title="c8">d9</font>,<font color="Olive" title="b1">x</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b9</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d9</font> )</span><br/>

<a NAME="E6:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A5</font></span>: 
 <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c7">c9</font>,<font color="Maroon" title="c8">d9</font>,2
 
;<br/>
<a NAME="E7:23"/>
 <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c7">c9</font>,<font color="Maroon" title="c8">d9</font>,1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E8">A4</font></span></a></span>;<br/>
<a NAME="E8:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c7">c9</font>,<font color="Maroon" title="c8">d9</font>,1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c9</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">d9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">b9</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">c9</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">d9</font>

;<br/>
<a NAME="E10:23"/>
(  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c7">c9</font>,<font color="Maroon" title="c8">d9</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K7" title="POLYEQ_1:func.7">Polynom</a> <font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c6">b9</font>,<font color="Maroon" title="c7">c9</font>,<font color="Maroon" title="c8">d9</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E8">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:23"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b9</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E6">A2</font></span></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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