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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">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</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>let </b><font color="Maroon" title="c5">i</font> be    <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/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">i0</font> = <font color="Maroon" title="c5">i</font> as    <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> <b>by </b><i><a class="txt" href="prvect_2.html#E1:35"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i0</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="c6">i0</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="c2">x</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E1:35"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="prvect_1.html#D10" title="PRVECT_1:def.10">PRVECT_1:def 10</a></i>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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>
<b>by </b><i><a class="ref" href="prvect_2.html#D4" target="_self" title="PRVECT_2:def.4">Def4</a></i>
<br/>.= 
 <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_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">i0</font> = <font color="Maroon" title="c5">i</font> as    <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> <b>by </b><i><a class="txt" href="prvect_2.html#E1:35"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E6:35"/><i><font color="Green" title="E28">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">x</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="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E7:35"/>
<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 href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c7">i0</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/>
<b>then reconsider </b><font color="Maroon" title="c8">xi0</font> = <font color="Maroon" title="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>, <font color="Maroon" title="c9">yi0</font> = <font color="Maroon" title="c3">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>, <font color="Maroon" title="c10">zi0</font> = <font color="Maroon" title="c4">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</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="c7">i0</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E1:35"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="prvect_2.html#E6:35"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E9:35"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c10">zi0</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c8">xi0</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">yi0</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#E3:35"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="prvect_2.html#D5" target="_self" title="PRVECT_2:def.5">Def5</a></i>;<br/>
<a NAME="E10:35"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c10">zi0</font></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"><font color="Maroon" title="c8">xi0</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c9">yi0</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D2" title="NORMSP_1:def.2">NORMSP_1:def 2</a></i>;<br/>
<a NAME="E11:35"/><i><font color="Green" title="E30">A5</font></i>: 
( the <a href="normsp_1.html#U1" title="NORMSP_1:sel.1">norm</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="c7">i0</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">yi0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; the <a href="normsp_1.html#U1" title="NORMSP_1:sel.1">norm</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="c7">i0</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">zi0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> )
 
<b>by </b><i><a class="ref" href="prvect_2.html#D11" target="_self" title="PRVECT_2:def.11">Def11</a></i>;<br/>
<a NAME="E12:35"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i0</font></span>)</span> <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#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i0</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T27" title="RVSUM_1:th.27">RVSUM_1:27</a></i>;<br/>
<b>hence </b><a NAME="E13:35"/>
<span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="prvect_2.html#E10:35"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="prvect_2.html#E11:35"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="prvect_2.html#D11" target="_self" title="PRVECT_2:def.11">Def11</a></i>;<br/>


</div>
