<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">z2</font>, <font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  &amp; ( <font color="Maroon" title="c2">z1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> or <font color="Maroon" title="c1">z2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> ) implies ( <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> &amp; <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">z3</font> ) )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">z29</font> = <font color="Maroon" title="c1">z2</font> as    <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">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" target="_self" title="COMPLFLD:def.1">Def1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:51"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c1">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c2">z1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> &amp; not <font color="Maroon" title="c1">z2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> ) or ( <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> &amp; <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">z3</font> ) )</span><br/>

<a NAME="E3:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z29</font> <a href="binop_2.html#K2" title="BINOP_2:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#T7" target="_self" title="COMPLFLD:th.7">Th7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:51"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
( <font color="Maroon" title="c2">z1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> or <font color="Maroon" title="c1">z2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> &amp; <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">z3</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1"/>
( <font color="Maroon" title="c2">z1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> or <font color="Maroon" title="c1">z2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E11">A3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_1_1"/>
<font color="Maroon" title="c2">z1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> &amp; <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">z3</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_1_1"/>
( <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> &amp; <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">z3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="complfld.html#T9" target="_self" title="COMPLFLD:th.9">Th9</a>, <a class="ref" href="xcmplx_1.html#T204" title="XCMPLX_1:th.204">XCMPLX_1:204</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_1_2"/>
<font color="Maroon" title="c1">z2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> &amp; <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">z3</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_1_2"/>
( <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z3</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> &amp; <font color="Maroon" title="c2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z2</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">z3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="complfld.html#T9" target="_self" title="COMPLFLD:th.9">Th9</a>, <a class="ref" href="xcmplx_1.html#T204" title="XCMPLX_1:th.204">XCMPLX_1:204</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
