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<b>let </b><font color="Maroon">z1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ;<br/><b>let </b><font color="Maroon">z2</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ;<br/>



<b>reconsider </b><font color="Maroon">z1'</font> = <font color="Maroon">z1</font>, <font color="Maroon">z2'</font> = <font color="Maroon">z2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:69"/>
( <font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z2</font> implies 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complfld.html#K3" target="_self">.|</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z2</font>
 ;<br/>

<a NAME="E2:69_1"/>
<font color="Maroon">z1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1'</font> <a href="binop_2.html#K4">-</a> <font color="Maroon">z2'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:69_1"/><b>then </b>
<span class="p1"><a href="complfld.html#K3" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complfld.html#K3" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1'</font> <a href="binop_2.html#K4">-</a> <font color="Maroon">z2'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
;<br/>
<b>hence </b><a NAME="E4:69_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complfld.html#K3" target="_self">.|</a></span>
 <b>by </b><i>, <a class="ref" href="complex1.html#T148">COMPLEX1:148</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:69"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complfld.html#K3" target="_self">.|</a></span>
 ;<br/>

<a NAME="E4:69"/>
<font color="Maroon">z1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1'</font> <a href="binop_2.html#K4">-</a> <font color="Maroon">z2'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:69"/><b>then </b>
<span class="p1"><a href="complfld.html#K3" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complfld.html#K3" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1'</font> <a href="binop_2.html#K4">-</a> <font color="Maroon">z2'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
;<br/>
<b>hence </b><a NAME="E6:69"/>
<font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z2</font>
 <b>by </b><i>, <a class="ref" href="complex1.html#T148">COMPLEX1:148</a></i>;<br/>


</div>
