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<span class="kw">let </span><font color="Maroon" title="c1">cF</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <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">  for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B1</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Olive" title="b2">B2</font> holds <br/> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">B2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Olive" title="b1">B1</font>)</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Olive" title="b2">B2</font>)</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">B1</font>, <font color="Maroon" title="c3">B2</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="membered.html#V6" title="MEMBERED:attr.6">natural-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="c2">B1</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c3">B2</font> implies  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Maroon" title="c2">B1</font>)</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Maroon" title="c3">B2</font>)</span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:106"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
<font color="Maroon" title="c2">B1</font> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c3">B2</font>
 ; <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> <span class="p1">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Maroon" title="c2">B1</font>)</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Maroon" title="c3">B2</font>)</span>)</span></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">B12</font> = <font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">B12L</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B1L</font> = <font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">B2L</font> = <font color="Maroon" title="c3">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span>;<br/>
<a NAME="E2:106"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>;<br/>
<a NAME="E3:106"/>
<font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span> <a href="afinsq_2.html#R1" title="AFINSQ_2:pred.1">&lt;N&lt;</a> <font color="Maroon" title="c3">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="afinsq_2.html#T25" target="_self" title="AFINSQ_2:th.25">Th36</a></span>;<br/>
<a NAME="E4:106"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A3</font></span>: 
 <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T35" target="_self" title="AFINSQ_2:th.35">Th46</a>, <a class="txt" href="#E2:106"><span class="lab"><font color="Green" title="E64">A2</font></span></a></span>;<br/>
<a NAME="E5:106"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D4" target="_self" title="AFINSQ_2:def.4">Def5</a></span>;<br/>
<a NAME="E6:106"/><span class="kw">then </span>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">cF</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">cF</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E7:106"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Maroon" title="c2">B1</font>)</span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Maroon" title="c3">B2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B2</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:106"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="afinsq_2.html#T70" target="_self" title="AFINSQ_2:th.70">Th82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:106"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Maroon" title="c2">B1</font>)</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K5" title="AFINSQ_2:func.5">SubXFinS</a> (<font color="Maroon" title="c1">cF</font>,<font color="Maroon" title="c3">B2</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T55" target="_self" title="AFINSQ_2:th.55">Th67</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/>


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