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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</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">F</font> is  <a href="valued_0.html#V4" title="VALUED_0:attr.4">rational-valued</a>  implies  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c1">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:80"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c1">F</font> is  <a href="valued_0.html#V4" title="VALUED_0:attr.4">rational-valued</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c1">F</font></span><br/>

<a NAME="E2:80"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D4" title="VALUED_0:def.4">VALUED_0:def 4</a></span>;<br/>
<a NAME="E3:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c1">F</font> is  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E4:80"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="membered.html#T1" title="MEMBERED:th.1">MEMBERED:1</a></span>;<br/>
<a NAME="E5:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c1">F</font> is  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_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="E1:80_1"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_1_1"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c1">F</font></span><br/><div class="add"><span class="kw">hence </span><a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c1">F</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D8" target="_self" title="AFINSQ_2:def.8">Def9</a>, <a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="binop_2.html#T3" title="BINOP_2:th.3">BINOP_2:3</a></span>
<br/>.= 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D8" target="_self" title="AFINSQ_2:def.8">Def9</a>, <a class="txt" href="#E5:80"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E1:80_1_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="binop_2.html#T1" title="BINOP_2:th.1">BINOP_2:1</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_1_2"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c1">F</font></span><br/><div class="add"><a NAME="E2:80_1_2"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
 <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="membered.html#T1" title="MEMBERED:th.1">MEMBERED:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  implies ( <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  ) )</span><br/><span class="kw">assume </span><a NAME="E1:80_1_2_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c3">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  ;<br/><a NAME="E3:80_1_2_1"/>
<a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="binop_2.html#K15" title="BINOP_2:func.15">+</a> <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D15" title="BINOP_2:def.15">BINOP_2:def 15</a></span>;<br/><span class="kw">hence </span><a NAME="E4:80_1_2_1"/>
( <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) &amp; <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D3" title="BINOP_2:def.3">BINOP_2:def 3</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">hence </span><a NAME="E4:80_1_2"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K39" title="BINOP_2:func.39">addrat</a> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T47" target="_self" title="AFINSQ_2:th.47">Th59</a>, <a class="txt" href="#E1:80_1_2"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E2:80_1_2"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E52">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></div><span class="kw">end;</span></div>

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