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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</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">LS</font> being    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Olive" title="b1">LS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Olive" title="b1">LS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">LS</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">LS</font> be    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">S</font>; <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">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c2">LS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c2">LS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">LS</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c2">LS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies  <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c2">LS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">LS</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">v</font> )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E3:40"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c4">p</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T58" title="FINSEQ_4:th.58">FINSEQ_4:58</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E5:40"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">LS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="finseq_3.html#T97" title="FINSEQ_3:th.97">FINSEQ_3:97</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c2">LS</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">LS</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T34" title="FINSEQ_2:th.34">FINSEQ_2:34</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:40"/>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c2">LS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c2">LS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">LS</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span><br/>

<span class="kw">hence </span> <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c2">LS</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">LS</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c5">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="rlaffin1.html#T30" target="_self" title="RLAFFIN1:th.30">Th30</a></span>
<br/>.= 
<font color="Maroon" title="c2">LS</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="rvsum_1.html#T73" title="RVSUM_1:th.73">RVSUM_1:73</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/>


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