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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nbvectsp.html#K5" title="NBVECTSP:func.5">-BinaryVectSp</a></span>)</span> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/>  for <font color="Olive" title="b2">Suml</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a><br/>  for <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Olive" title="b2">Suml</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">L</font> &amp; <font color="Olive" title="b3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/> ex <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b2">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b4">x</font> &amp; (  for <font color="Olive" title="b5">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b6">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b3">j</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">L</font> be   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nbvectsp.html#K5" title="NBVECTSP:func.5">-BinaryVectSp</a></span>)</span> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Suml</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a><br/>  for <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Olive" title="b1">Suml</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Olive" title="b2">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/> ex <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b1">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b3">x</font> &amp; (  for <font color="Olive" title="b4">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b5">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">Suml</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">Suml</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c4">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">x</font> &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b4">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">j</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">Suml</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c5">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c4">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">x</font> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b3">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> ) ) ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">Suml</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c5">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c4">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">x</font> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b3">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> ) ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a><span class="kw"> such that </span><br/><a NAME="E3:18"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b2">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#T6" target="_self" onmouseover="rs('nbvectsp/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">f</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nbvectsp.html#K5" title="NBVECTSP:func.5">-BinaryVectSp</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:18"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
(  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp; <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nbvectsp.html#K5" title="NBVECTSP:func.5">-BinaryVectSp</a></span>)</span> &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nbvectsp.html#K5" title="NBVECTSP:func.5">-BinaryVectSp</a></span>)</span>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D12" onmouseover="rs('rlvect_1/D12')" onmouseout="rh()">RLVECT_1:def 12</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> );<br/>
<a NAME="E6:18"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">K</font> = <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">i</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">y</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c10">y</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:18_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c10">y</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">consider </span><font color="Maroon" title="c8">g</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a><span class="kw"> such that </span><br/><a NAME="E8:18"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A4</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c9">Sumlj</font> = <font color="Maroon" title="c4">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font>;<br/>
<a NAME="E9:18"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c4">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">x</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:18_2"/>
 ex <font color="Olive" title="b1">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp; <font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:18_2"/>
<font color="Maroon" title="c4">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E10:18"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:18_3"/>
 ex <font color="Olive" title="b1">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> &amp; <font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:18_3"/>
<font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="bspace.html#T5" onmouseover="rs('bspace/T5')" onmouseout="rh()">BSPACE:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E11:18"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">vj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive" title="b2">vj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">vj</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">vj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  st <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive" title="b1">vj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b1">vj</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">vj</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c11">vj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">vj</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:18_4"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c11">vj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">vj</font></span><br/>

<a NAME="E2:18_4"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c6">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E3:18_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">LVn</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a><span class="kw"> such that </span><br/><a NAME="E5:18_4"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
( <font color="Maroon" title="c12">LVn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">LVn</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">Vn</font> = <font color="Maroon" title="c12">LVn</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nbvectsp.html#K5" title="NBVECTSP:func.5">-BinaryVectSp</a></span>)</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">K1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a><span class="kw"> such that </span><br/><a NAME="E8:18_4"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
( <font color="Maroon" title="c14">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">K1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">VK1</font> = <font color="Maroon" title="c14">K1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nbvectsp.html#K5" title="NBVECTSP:func.5">-BinaryVectSp</a></span>)</span> ;<br/>
<a NAME="E10:18_4"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font>]</span><br/>

<a NAME="E1:18_4_1"/>
<font color="Maroon" title="c16">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:18_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">i</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">then </span><span class="kw">consider </span><font color="Maroon" title="c16">K0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a><span class="kw"> such that </span><br/><a NAME="E12:18_4"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
( <font color="Maroon" title="c16">K0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">K0</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">VK0</font> = <font color="Maroon" title="c16">K0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nbvectsp.html#K5" title="NBVECTSP:func.5">-BinaryVectSp</a></span>)</span> ;<br/>
<a NAME="E14:18_4"/>
<font color="Maroon" title="c17">VK0</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">Vn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c16">K0</font>,<font color="Maroon" title="c12">LVn</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:18_4"/>
<font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">vj</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18_4"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E1:18_4"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E5:18_4"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E12:18_4"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="descip_1.html#D4" onmouseover="rs('descip_1/D4')" onmouseout="rh()">DESCIP_1:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E12:18"/>
<font color="Maroon" title="c4">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E11:18"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="rlvect_1.html#D12" onmouseover="rs('rlvect_1/D12')" onmouseout="rh()">RLVECT_1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:18"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c4">Suml</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">x</font> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b3">K</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c5">j</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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