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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>suppose </b></a><a NAME="E1:63_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#T50" target="_self">Th50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2"><b>suppose </b></a><a NAME="E1:63_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:63_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="binop_2.html#K7">-</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E4:63_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E85"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#T50" target="_self">Th50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="group_1.html#E85"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="group_1.html#E85"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#T19" target="_self">Th19</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_3"><b>suppose </b></a><a NAME="E1:63_1_3"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:63_1_3"/><b>then </b>
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="binop_2.html#K7">-</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E4:63_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E85"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#T50" target="_self">Th50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="group_1.html#E85"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#T19" target="_self">Th19</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_4"><b>suppose </b></a><a NAME="E1:63_1_4"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:63_1_4"/><b>then </b>
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="binop_2.html#K7">-</a> 0 &amp;  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="binop_2.html#K7">-</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E4:63_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#T50" target="_self">Th50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="group_1.html#E85"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="group_1.html#E85"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="group_1.html#E85"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#T19" target="_self">Th19</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
