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<b>let </b><font color="Maroon" title="c1">K</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">j</font> &amp; ( for <font color="Olive" title="b4">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">j</font> holds <br/><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">j</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> &amp; ( for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span></span><br/>



<div><i><font color="Green" title="E10">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) implies  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:9_1"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font> )
 <b>and </b><br/><a NAME="E2:9_1"/><i><font color="Green" title="E12">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span><br/><b>set </b><font color="Maroon" title="c5">pI</font> = <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font>;<br/><b>consider </b><font color="Maroon" title="c6">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E4:9_1"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T101" title="FINSEQ_1:th.101">FINSEQ_1:101</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c6">q</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="matrix15.html#E4:9_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/><a NAME="E6:9_1"/><i><font color="Green" title="E14">A5</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:9_1"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E8:9_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E6:9_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="matrix15.html#E1:9_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:9_1"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">i</font> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a>, <a class="txt" href="matrix15.html#E7:9_1"><i><font color="Green" title="E15">A6</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> &amp; <font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )</span><br/><b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> &amp; <font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span><br/><b>reconsider </b><font color="Maroon" title="c9">kk</font> = <font color="Maroon" title="c8">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:9_1_1"/><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:9_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="matrix15.html#E1:9_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> = 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">kk</font>
<b>by </b><i><a class="txt" href="matrix15.html#E4:9_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="matrix15.html#E1:9_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<b>by </b><i><a class="txt" href="matrix15.html#E6:9_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="matrix15.html#E2:9_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="matrix15.html#E1:9_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="matrix15.html#E3:9_1_1"><i><font color="Green" title="E18">A9</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then </b><i><font color="Green" title="E17">A10</font></i>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="matrix_3.html#T14" title="MATRIX_3:th.14">MATRIX_3:14</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="matrix15.html#E4:9_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="matrix15.html#E6:9_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><a NAME="E12:9_1"/>
not <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:9_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">ji</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:9_1"/><i><font color="Green" title="E18">A11</font></i>: 
( <font color="Maroon" title="c8">ji</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">ji</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="matrix15.html#E1:9_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="matrix15.html#E4:9_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">ji</font> implies <font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )</span><br/><b>assume </b><a NAME="E1:9_1_3"/><i><font color="Green" title="E19">A12</font></i>: 
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">ji</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span><br/><b>reconsider </b><font color="Maroon" title="c10">kk</font> = <font color="Maroon" title="c9">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:9_1_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:9_1_3"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="matrix15.html#E1:9_1_3"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:9_1_3"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E6:9_1_3"/><b>then </b><i><font color="Green" title="E20">A13</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">kk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">kk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">ji</font> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:9_1_3"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>thus </b><font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">kk</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E4:9_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="matrix15.html#E1:9_1_3"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
<b>by </b><i><a class="txt" href="matrix15.html#E2:9_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="matrix15.html#E14:9_1"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="matrix15.html#E6:9_1_3"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="matrix15.html#E4:9_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="matrix15.html#E1:9_1_3"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then </b> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c7">q</font> = 
<font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">ji</font>
<b>by </b><i><a class="txt" href="matrix15.html#E14:9_1"><i><font color="Green" title="E18">A11</font></i></a>, <a class="ref" href="matrix_3.html#T14" title="MATRIX_3:th.14">MATRIX_3:14</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="matrix15.html#E14:9_1"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="matrix15.html#E9:9_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="matrix15.html#E4:9_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="matrix15.html#E1:9_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><b>hence </b><a NAME="E17:9_1"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E11:9_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="matrix15.html#E4:9_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) implies  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E2:9"/><i><font color="Green" title="E11">A14</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> )
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green" title="E12">A15</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span><br/>

<a NAME="E4:9"/><i><font color="Green" title="E13">A16</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 
<b>by </b><i><a class="txt" href="matrix15.html#E3:9"><i><font color="Green" title="E12">A15</font></i></a></i>;<br/>
<a NAME="E5:9"/>
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font> or <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> )
 
<b>by </b><i><a class="txt" href="matrix15.html#E2:9"><i><font color="Green" title="E11">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:9"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="matrix15.html#E2:9"><i><font color="Green" title="E11">A14</font></i></a>, <a class="txt" href="matrix15.html#E3:9"><i><font color="Green" title="E12">A15</font></i></a>, <a class="txt" href="matrix15.html#E4:9"><i><font color="Green" title="E13">A16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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