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<div class="add">

<a NAME="E1:27_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">F</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T11" target="_self" title="PRVECT_1:th.11">Th11</a></i>;<br/>
<a NAME="E2:27_1_1"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>
 
<b>by </b><i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></i>;<br/>
<a NAME="E3:27_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="#E1:27_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>
<a NAME="E4:27_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 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="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,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="c2">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K5" title="PRVECT_1:func.5">-Group_over</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>;<br/>
<a NAME="E5:27_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K5" title="PRVECT_1:func.5">-Group_over</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">.--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="prvect_1.html#D3" target="_self" title="PRVECT_1:def.3">Def3</a></i>;<br/>
<a NAME="E6:27_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font> is <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">VectSp-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font></span>)</span> holds <br/> ( <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font> be   <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="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c6">w</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c7">v'</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c8">w'</font> = <font color="Maroon" title="c6">w</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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> <b>by </b><i><a class="txt" href="#E4:27_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E5:27_1_1"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:27_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E5:27_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <font color="Maroon" title="c7">v'</font>,<font color="Maroon" title="c8">w'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <font color="Maroon" title="c7">v'</font>,<font color="Maroon" title="c8">w'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">v'</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c8">w'</font></span>)</span>
<b>by </b><i><a class="txt" href="prvect_1.html#E16"><i><font color="Green" title="E8">Lm5</font></i></a>, <a class="ref" href="prvect_1.html#T18" target="_self" title="PRVECT_1:th.18">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">v'</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c8">w'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">v'</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c8">w'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">v'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c8">w'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">v</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c8">w'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c6">w</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:27_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E5:27_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
; <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="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> )</span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c7">v'</font>
<b>by </b><i><a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c7">v'</font>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">v'</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="txt" href="prvect_1.html#E16"><i><font color="Green" title="E8">Lm5</font></i></a>, <a class="ref" href="finseqop.html#T47" title="FINSEQOP:th.47">FINSEQOP:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">v'</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">v'</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">v'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">v</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:27_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E5:27_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
; <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="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> )</span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c7">v'</font>
<b>by </b><i><a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c7">v'</font>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="txt" href="prvect_1.html#E9"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="finseqop.html#T32" title="FINSEQOP:th.32">FINSEQOP:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
; <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"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font></span><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-Mult_over</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c7">v'</font>
<b>by </b><i><a class="ref" href="prvect_1.html#D5" target="_self" title="PRVECT_1:def.5">Def5</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">[;]</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c7">v'</font>
<b>by </b><i><a class="ref" href="prvect_1.html#D4" target="_self" title="PRVECT_1:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="txt" href="#E2:27_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="finseqop.html#T58" title="FINSEQOP:th.58">FINSEQOP: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/>

<b>thus </b><a NAME="E6:27_1_1_1"/>
verum
 ; <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>hence </b><a NAME="E7:27_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">F</font> is <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">VectSp-like</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
