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<span class="kw">set </span><font color="Maroon" title="c1">x</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">a3</font>, <font color="Maroon" title="c5">a4</font>, <font color="Maroon" title="c6">a5</font>, <font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c8">b2</font>, <font color="Maroon" title="c9">b3</font>, <font color="Maroon" title="c10">b4</font>, <font color="Maroon" title="c11">b5</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="firebrick">::  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_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c6">a5</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>,<font color="Maroon" title="c11">b5</font>,<font color="Olive" title="b1">x</font>) ) implies ( <font color="Maroon" title="c6">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b5</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">a2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">a3</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">b2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">b3</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">b4</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">a3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">b2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">b3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">b4</font> ) )</span><br/>

<a NAME="E1:10"/><span class="lab"><font color="Green" title="E7">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; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<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="kw">assume </span><a NAME="E2:10"/><span class="lab"><font color="Green" title="E8">A2</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_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c6">a5</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>,<font color="Maroon" title="c11">b5</font>,<font color="Olive" title="b1">x</font>)
 ; <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="c6">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b5</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">a2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">a3</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">b2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">b3</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">b4</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">a3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">b2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">b3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">b4</font> )</span><br/>

<a NAME="E3:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 <a href="polyeq_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c6">a5</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>,<font color="Maroon" title="c11">b5</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>)
 
;<br/>
<a NAME="E4:10"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
( 1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/>
<a NAME="E5:10"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 4 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyeq_2.html#T4" target="_self" title="POLYEQ_2:th.4">Th4</a></span>;<br/>
<a NAME="E6:10"/>
(  <a href="polyeq_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c6">a5</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_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>,<font color="Maroon" title="c11">b5</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) &amp;  <a href="polyeq_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font>,<font color="Maroon" title="c6">a5</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_2.html#K1" title="POLYEQ_2:func.1">Polynom</a> (<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font>,<font color="Maroon" title="c9">b3</font>,<font color="Maroon" title="c10">b4</font>,<font color="Maroon" title="c11">b5</font>,1) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:10"/>
( <font color="Maroon" title="c6">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b5</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">a2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">a3</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">b2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">b3</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">b4</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">a3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">b2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">b3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">b4</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E11">A5</font></span></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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