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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</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"> <font color="Maroon" title="c1">A</font> <a href="member_1.html#K11" title="MEMBER_1:func.11">--</a> <font color="Maroon" title="c2">B</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="Olive" title="b1">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Olive" title="b2">c2</font></span>)</span> where <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> is    <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>  : ( <font color="Olive" title="b1">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span> } </span> </span><br/>



<span class="kw">thus </span><a NAME="E1:108"/>
<font color="Maroon" title="c1">A</font> <a href="member_1.html#K11" title="MEMBER_1:func.11">--</a> <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Olive" title="b2">c2</font></span>)</span> where <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> is    <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>  : ( <font color="Olive" title="b1">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span> } </span> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Olive" title="b2">c2</font></span>)</span> where <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> is    <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>  : ( <font color="Olive" title="b1">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> <a href="member_1.html#K11" title="MEMBER_1:func.11">--</a> <font color="Maroon" title="c2">B</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="member_1.html#K11" title="MEMBER_1:func.11">--</a> <font color="Maroon" title="c2">B</font> or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Olive" title="b2">c2</font></span>)</span> where <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> is    <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>  : ( <font color="Olive" title="b1">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:108_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="member_1.html#K11" title="MEMBER_1:func.11">--</a> <font color="Maroon" title="c2">B</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Olive" title="b2">c2</font></span>)</span> where <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> is    <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>  : ( <font color="Olive" title="b1">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">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="E3:108_1"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c1</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c5">c2</font>
 <span class="kw">and </span><br/><a NAME="E4:108_1"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c4">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:108_1"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<font color="Maroon" title="c5">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="member_1.html#K5" title="MEMBER_1:func.5">--</a> <font color="Maroon" title="c2">B</font>
 ;<br/>
<a NAME="E6:108_1"/>
( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <font color="Maroon" title="c5">c2</font></span>)</span> &amp;  <a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <font color="Maroon" title="c5">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E5:108_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="member_1.html#T12" target="_self" title="MEMBER_1:th.12">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:108_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Olive" title="b2">c2</font></span>)</span> where <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> is    <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>  : ( <font color="Olive" title="b1">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108_1"><span class="lab"><font color="Green" title="E57">A2</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/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Olive" title="b2">c2</font></span>)</span> where <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> is    <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>  : ( <font color="Olive" title="b1">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) </span> } </span>  or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="member_1.html#K11" title="MEMBER_1:func.11">--</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:108"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Olive" title="b2">c2</font></span>)</span> where <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> is    <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>  : ( <font color="Olive" title="b1">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="member_1.html#K11" title="MEMBER_1:func.11">--</a> <font color="Maroon" title="c2">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">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="E4:108"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c1</font> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <font color="Maroon" title="c5">c2</font> &amp; <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <span class="kw">and </span><br/><a NAME="E5:108"/><span class="lab"><font color="Green" title="E57">A5</font></span>: 
<font color="Maroon" title="c5">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ;<br/>
<a NAME="E6:108"/>
 <a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <font color="Maroon" title="c5">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="member_1.html#K5" title="MEMBER_1:func.5">--</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E57">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:108"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="member_1.html#K11" title="MEMBER_1:func.11">--</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E56">A4</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/>


</div>
