<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="prvect_2.html#NM2" title="PRVECT_2:NM.2">RealLinearSpace-Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">w1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span><br/> for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">w1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b3">i</font></span>)</span> &amp; ( for <font color="Olive" title="b4">vi</font>, <font color="Olive" title="b5">wi</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Olive" title="b3">i</font></span>)</span>  st <font color="Olive" title="b4">vi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">wi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b3">i</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">vi</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b5">wi</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">w1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">w1</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; ( for <font color="Olive" title="b2">vi</font>, <font color="Olive" title="b3">wi</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span>  st <font color="Olive" title="b2">vi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">wi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">w1</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">vi</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">wi</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">w1</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> &amp; ( for <font color="Olive" title="b1">vi</font>, <font color="Olive" title="b2">wi</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c4">i</font></span>)</span>  st <font color="Olive" title="b1">vi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">wi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">w1</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">vi</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">wi</font> ) )</span><br/>





<a NAME="E1:20"/>
<span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">w1</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D10" title="PRVECT_1:def.10">PRVECT_1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:20"/>
( <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">w1</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">w1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> &amp; ( for <font color="Olive" title="b1">vi</font>, <font color="Olive" title="b2">wi</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c4">i</font></span>)</span>  st <font color="Olive" title="b1">vi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">wi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w1</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">w1</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">vi</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">wi</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D5" target="_self" title="PRVECT_2:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
