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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">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> <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">xi</font>, <font color="Maroon" title="c6">yi</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c2">i</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">zx</font>, <font color="Maroon" title="c8">zy</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>;<br/>









<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c5">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">zx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c7">zx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c6">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">zy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c8">zy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">I</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:41"/><i><font color="Green" title="E32">A2</font></i>: 
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#G1" title="NORMSP_1:aggr.1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<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></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K8" title="PRVECT_2:func.8">zeros</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><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></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>,<span class="p1"><a href="prvect_2.html#K2" title="PRVECT_2:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K9" title="PRVECT_2:func.9">multop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_2.html#K2" title="PRVECT_2:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K13" title="PRVECT_2:func.13">productnorm</a> <font color="Maroon" title="c1">G</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="prvect_2.html#T6" target="_self" title="PRVECT_2:th.6">Th6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">mzx</font> =  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x</font> as    <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/>
<b>reconsider </b><font color="Maroon" title="c11">zyPG</font> = <font color="Maroon" title="c8">zy</font>, <font color="Maroon" title="c12">zxPG</font> = <font color="Maroon" title="c7">zx</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E3:41"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">zyi</font> = <font color="Maroon" title="c8">zy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c14">zxi</font> = <font color="Maroon" title="c7">zx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</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="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E1:41"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/>
<a NAME="E7:41"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>
 
<b>by </b><i><a class="ref" href="prvect_2.html#D4" target="_self" title="PRVECT_2:def.4">Def4</a></i>;<br/>
<a NAME="E8:41"/><b>then </b><i><font color="Green" title="E33">A3</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:41"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></i>;<br/>
<a NAME="E10:41"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c10">mzx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c14">zxi</font>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E1:41"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="prvect_2.html#E3:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="prvect_2.html#E8:41"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="prvect_2.html#E18"><i><font color="Green" title="E16">Lm4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">mzxi</font> = <font color="Maroon" title="c10">mzx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</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="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c2">i</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">zyMm</font> = <font color="Maroon" title="c4">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">x</font> as    <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> <b>by </b><i><a class="txt" href="prvect_2.html#E3:41"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<a NAME="E13:41"/>
<font color="Maroon" title="c16">zyMm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">zyi</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">mzxi</font>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E1:41"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="prvect_2.html#E3:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="prvect_2.html#E8:41"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="prvect_2.html#E17"><i><font color="Green" title="E15">Lm3</font></i></a></i>;<br/>
<a NAME="E14:41"/><b>then </b>
<font color="Maroon" title="c16">zyMm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">zyi</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c14">zxi</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E10:41"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E15:41"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">yi</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">xi</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="txt" href="prvect_2.html#E1:41"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="prvect_2.html#T10" target="_self" title="PRVECT_2:th.10">Th10</a></i>;<br/>


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