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<span class="kw">let </span><font color="Maroon" title="c1">B</font>, <font color="Maroon" title="c2">C</font> be   <a href="membered.html#V1" title="MEMBERED:attr.1">complex-membered</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">a</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  <font color="Olive" title="b1">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c2">C</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c2">C</font></span>)</span></span><br/>





<a NAME="E1:317"/><span class="lab"><font color="Green" title="E134">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="317_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">z</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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="membered.html#D7" title="MEMBERED:def.7">MEMBERED:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c2">C</font></span>)</span> or <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">C</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:317_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c2">C</font></span>)</span>
 ; <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="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">C</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">c1</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> <span class="kw"> such that </span><br/><a NAME="E3:317_1"/><span class="lab"><font color="Green" title="E135">A2</font></span>: 
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c6">c1</font>
 <span class="kw">and </span><br/><a NAME="E4:317_1"/><span class="lab"><font color="Green" title="E136">A3</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c1">B</font>
 <span class="kw">and </span><br/><a NAME="E5:317_1"/><span class="lab"><font color="Green" title="E137">A4</font></span>: 
<font color="Maroon" title="c6">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c2">C</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">c3</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> <span class="kw"> such that </span><br/><a NAME="E7:317_1"/><span class="lab"><font color="Green" title="E138">A5</font></span>: 
<font color="Maroon" title="c6">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c7">c3</font>
 <span class="kw">and </span><br/><a NAME="E8:317_1"/><span class="lab"><font color="Green" title="E139">A6</font></span>: 
<font color="Maroon" title="c7">c3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:317_1"><span class="lab"><font color="Green" title="E137">A4</font></span></a>, <a class="ref" href="member_1.html#T195" target="_self" title="MEMBER_1:th.195">Th195</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">c2</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> <span class="kw"> such that </span><br/><a NAME="E10:317_1"/><span class="lab"><font color="Green" title="E140">A7</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c8">c2</font>
 <span class="kw">and </span><br/><a NAME="E11:317_1"/><span class="lab"><font color="Green" title="E141">A8</font></span>: 
<font color="Maroon" title="c8">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:317_1"><span class="lab"><font color="Green" title="E136">A3</font></span></a>, <a class="ref" href="member_1.html#T195" target="_self" title="MEMBER_1:th.195">Th195</a></span>;<br/>
<a NAME="E12:317_1"/><span class="lab"><font color="Green" title="E142">A9</font></span>: 
<font color="Maroon" title="c8">c2</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c7">c3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:317_1"><span class="lab"><font color="Green" title="E141">A8</font></span></a>, <a class="txt" href="#E8:317_1"><span class="lab"><font color="Green" title="E139">A6</font></span></a></span>;<br/>
<a NAME="E13:317_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c7">c3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:317_1"><span class="lab"><font color="Green" title="E135">A2</font></span></a>, <a class="txt" href="#E10:317_1"><span class="lab"><font color="Green" title="E140">A7</font></span></a>, <a class="txt" href="#E7:317_1"><span class="lab"><font color="Green" title="E138">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:317_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:317_1"><span class="lab"><font color="Green" title="E142">A9</font></span></a>, <a class="ref" href="member_1.html#T193" target="_self" title="MEMBER_1:th.193">Th193</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/>


</div><span class="kw">end;</span></div>
<a NAME="E2:317"/>
<font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="member_1.html#T95" target="_self" title="MEMBER_1:th.95">Th95</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:317"/>
<font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="member_1.html#K23" title="MEMBER_1:func.23">**</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:317"><span class="lab"><font color="Green" title="E134">A1</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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/>


</div>
