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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">z</font>, <font color="Olive" title="b2">z1</font>, <font color="Olive" title="b3">z2</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b4">z</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="polyeq_1.html#K3" title="POLYEQ_1:func.3">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="Olive" title="b4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K5" title="POLYEQ_1:func.5">Quard</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">z1</font>,<font color="Olive" title="b3">z2</font>,<font color="Olive" title="b4">z</font> ) holds <br/>( <font color="Maroon" title="c2">b</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">z2</font></span>)</span> &amp; <font color="Maroon" title="c3">c</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">z2</font> )</span><br/><b>let </b><font color="Maroon" title="c4">z</font>, <font color="Maroon" title="c5">z1</font>, <font color="Maroon" title="c6">z2</font> be  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</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="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">z</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="polyeq_1.html#K3" title="POLYEQ_1:func.3">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="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K5" title="POLYEQ_1:func.5">Quard</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c6">z2</font>,<font color="Olive" title="b1">z</font> ) implies ( <font color="Maroon" title="c2">b</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">z2</font></span>)</span> &amp; <font color="Maroon" title="c3">c</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">z2</font> ) )</span><br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> (  ex <font color="Olive" title="b1">z</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st not  <a href="polyeq_1.html#K3" title="POLYEQ_1:func.3">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="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K5" title="POLYEQ_1:func.5">Quard</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c6">z2</font>,<font color="Olive" title="b1">z</font> or ( <font color="Maroon" title="c2">b</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">z2</font></span>)</span> &amp; <font color="Maroon" title="c3">c</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">z2</font> ) )</span><br/>

<b>assume </b><a NAME="E2:7"/><i><font color="Green" title="E5">A2</font></i>: 
for <font color="Olive" title="b1">z</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="polyeq_1.html#K3" title="POLYEQ_1:func.3">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="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyeq_1.html#K5" title="POLYEQ_1:func.5">Quard</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c6">z2</font>,<font color="Olive" title="b1">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">z2</font></span>)</span> &amp; <font color="Maroon" title="c3">c</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">z2</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c7">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>
<a NAME="E4:7"/><i><font color="Green" title="E6">A3</font></i>: 
 <a href="polyeq_1.html#K4" title="POLYEQ_1:func.4">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</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#K5" title="POLYEQ_1:func.5">Quard</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c6">z2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="polyeq_3.html#E2:7"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
 <a href="polyeq_1.html#K5" title="POLYEQ_1:func.5">Quard</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c6">z2</font>,1 = 
 <a href="polyeq_1.html#K4" title="POLYEQ_1:func.4">Polynom</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,1
<b>by </b><i><a class="txt" href="polyeq_3.html#E2:7"><i><font color="Green" title="E5">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>

;<br/>
<a NAME="E6:7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="txt" href="polyeq_3.html#E4:7"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:7"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">c</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">z2</font> )
 <b>by </b><i><a class="txt" href="polyeq_3.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="polyeq_3.html#E4:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T204" title="XCMPLX_1:th.204">XCMPLX_1:204</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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