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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">I</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b2">x</font> being   <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">X</font></span>)</span><br/>  for <font color="Olive" title="b3">y</font> being   <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="c2">Y</font></span>)</span>  ex <font color="Olive" title="b4">x1</font>, <font color="Olive" title="b5">y1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x1</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">y1</font> &amp; <font color="Olive" title="b1">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b5">y1</font> ) ) &amp; (  for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">w</font></span>)</span> ) &amp; (  for <font color="Olive" title="b2">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span><br/>  for <font color="Olive" title="b3">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span></span>)</span> <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"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b2">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b2">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">X0</font> = <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c4">Y0</font> = <font color="Maroon" title="c2">Y</font> as   <a href="prvect_2.html#NM2" title="PRVECT_2:NM.2">RealLinearSpace-Sequence</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">PX</font> =  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">PY</font> =  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">PX0</font> =  <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">PY0</font> =  <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">CX</font> =  <a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c10">CY</font> =  <a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c2">Y</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">CX0</font> =  <a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c3">X0</font>, <font color="Maroon" title="c12">CY0</font> =  <a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c4">Y0</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E4:46"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
(  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</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">X</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">X</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">X</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">X</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">X</font></span>)</span> #) &amp;  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</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="c2">Y</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="c2">Y</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="c2">Y</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="c2">Y</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="c2">Y</font></span>)</span> #) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#T6" title="PRVECT_2:th.6">PRVECT_2:6</a></span>;<br/>
<a NAME="E5:46"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <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">X</font></span>)</span><br/>  for <font color="Olive" title="b3">f1</font>, <font color="Olive" title="b4">f2</font> being   <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="c2">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="prvect_3.html#K1" title="PRVECT_3:func.1">prod_ADD</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">g1</font>,<font color="Olive" title="b3">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g2</font>,<font color="Olive" title="b4">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">g2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">f2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">g1</font>, <font color="Maroon" title="c14">g2</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">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <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="c2">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="prvect_3.html#K1" title="PRVECT_3:func.1">prod_ADD</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">g1</font>,<font color="Olive" title="b1">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g2</font>,<font color="Olive" title="b2">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">g2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">f2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c15">f1</font>, <font color="Maroon" title="c16">f2</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="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_3.html#K1" title="PRVECT_3:func.1">prod_ADD</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">g1</font>,<font color="Maroon" title="c15">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g2</font>,<font color="Maroon" title="c16">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">g2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">f2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c17">g10</font> = <font color="Maroon" title="c13">g1</font>, <font color="Maroon" title="c18">g20</font> = <font color="Maroon" title="c14">g2</font> as   <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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">f10</font> = <font color="Maroon" title="c15">f1</font>, <font color="Maroon" title="c20">f20</font> = <font color="Maroon" title="c16">f2</font> as   <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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E3:46_1"/>
( <font color="Maroon" title="c17">g10</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">g20</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">g2</font> &amp; <font color="Maroon" title="c19">f10</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">f20</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">f2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:46_1"/>
<span class="p1">(<span class="default"><a href="prvect_3.html#K1" title="PRVECT_3:func.1">prod_ADD</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">g1</font>,<font color="Maroon" title="c15">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g2</font>,<font color="Maroon" title="c16">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">g1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">g2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">f2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D1" target="_self" title="PRVECT_3:def.1">Def1</a></span>; <a class="txt" 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="E6:46"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 for <font color="Olive" title="b1">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/>  for <font color="Olive" title="b2">g</font> being   <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">X</font></span>)</span><br/>  for <font color="Olive" title="b3">f</font> being   <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="c2">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="prvect_3.html#K2" title="PRVECT_3:func.2">prod_MLT</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">r</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">f</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <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">X</font></span>)</span><br/>  for <font color="Olive" title="b2">f</font> being   <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="c2">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="prvect_3.html#K2" title="PRVECT_3:func.2">prod_MLT</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">r</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">f</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c14">g</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">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <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="c2">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="prvect_3.html#K2" title="PRVECT_3:func.2">prod_MLT</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">r</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c15">f</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="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_3.html#K2" title="PRVECT_3:func.2">prod_MLT</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">r</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">f</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c16">g0</font> = <font color="Maroon" title="c14">g</font> as   <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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">f0</font> = <font color="Maroon" title="c15">f</font> as   <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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E3:46_2"/>
( <font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">g0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">g</font> &amp; <font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:46_2"/>
<span class="p1">(<span class="default"><a href="prvect_3.html#K2" title="PRVECT_3:func.2">prod_MLT</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">r</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">f</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D2" target="_self" title="PRVECT_3:def.2">Def2</a></span>; <a class="txt" 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="E7:46"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
(  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp;  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">Y0</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c3">X0</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">Y0</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">CX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">CY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">CX0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">X0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">CY0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Y0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D4" title="PRVECT_2:def.4">PRVECT_2:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">I0</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">Y0</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:46"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
( <font color="Maroon" title="c13">I0</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c13">I0</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b1">x</font> being   <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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span><br/>  for <font color="Olive" title="b2">y</font> being   <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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span>  ex <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">y1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y1</font> &amp; <font color="Maroon" title="c13">I0</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b4">y1</font> ) ) &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">:]</a></span> holds  <font color="Maroon" title="c13">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">:]</a></span><br/>  for <font color="Olive" title="b2">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c13">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">Y0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p3">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">:]</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T13" target="_self" title="PRVECT_3:th.13">Th13</a></span>;<br/>
<a NAME="E10:46"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span> <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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K8" title="PRVECT_2:func.8">zeros</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#T6" title="PRVECT_2:th.6">PRVECT_2:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">I</font> = <font color="Maroon" title="c13">I0</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">I</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c14">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b1">x</font> being   <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">X</font></span>)</span><br/>  for <font color="Olive" title="b2">y</font> being   <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="c2">Y</font></span>)</span>  ex <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">y1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y1</font> &amp; <font color="Maroon" title="c14">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b4">y1</font> ) ) &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span><br/>  for <font color="Olive" title="b2">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> ) &amp; <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span></span>)</span> <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"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E12:46"/>
( <font color="Maroon" title="c14">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c14">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font> being   <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">X</font></span>)</span><br/>  for <font color="Olive" title="b2">y</font> being   <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="c2">Y</font></span>)</span>  ex <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">y1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y1</font> &amp; <font color="Maroon" title="c14">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b4">y1</font> ) ) &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span><br/>  for <font color="Olive" title="b2">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> ) &amp; <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span></span>)</span> <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"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:46"/>
 for <font color="Olive" title="b1">x</font> being   <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">X</font></span>)</span><br/>  for <font color="Olive" title="b2">y</font> being   <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="c2">Y</font></span>)</span>  ex <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">y1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y1</font> &amp; <font color="Maroon" title="c14">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b4">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span><br/>  for <font color="Olive" title="b2">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> ) &amp; <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span></span>)</span> <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"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> ) )</span><br/>

<a NAME="E14:46"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">X</font></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">Y</font></span>)</span> holds <br/><font color="Maroon" title="c14">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">X</font></span>)</span> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">Y</font></span>)</span> implies <font color="Maroon" title="c14">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_3"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">X</font></span>)</span> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">Y</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">y</font></span><br/>

<a NAME="E2:46_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y1</font> &amp; <font color="Maroon" title="c14">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:46_3"/>
<font color="Maroon" title="c14">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">y</font>
 ; <a class="txt" 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">thus </span><a NAME="E15:46"/>
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span><br/>  for <font color="Olive" title="b2">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> ) &amp; <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span></span>)</span> <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"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">v</font>, <font color="Maroon" title="c16">w</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c17">v0</font> = <font color="Maroon" title="c15">v</font>, <font color="Maroon" title="c18">w0</font> = <font color="Maroon" title="c16">w</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E2:46_4"/>
<font color="Maroon" title="c15">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v0</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">w0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="prvect_3.html#D1" target="_self" title="PRVECT_3:def.1">Def1</a></span>;<br/>
<a NAME="E3:46_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">v0</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">w0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:46_4"/>
<font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>; <a class="txt" 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">thus </span><a NAME="E16:46"/>
 for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span><br/>  for <font color="Olive" title="b2">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span></span>)</span> <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"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">v</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c16">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c17">v0</font> = <font color="Maroon" title="c15">v</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">X0</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c4">Y0</font></span>)</span></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E2:46_5"/>
<font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">v0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="prvect_3.html#D2" target="_self" title="PRVECT_3:def.2">Def2</a></span>;<br/>
<a NAME="E3:46_5"/><span class="kw">then </span>
<font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">v0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:46_5"/>
<font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>; <a class="txt" 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">thus </span><a NAME="E17:46"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/>

<a NAME="E18:46"/>
 for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">v</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c15">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c16">x1</font> being   <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">X</font></span>)</span>, <font color="Maroon" title="c17">y1</font> being   <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="c2">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:46_6"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="prvect_3.html#E7"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">v1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<span class="kw"> such that </span><br/><a NAME="E4:46_6"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
( <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c16">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>,<span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="prvect_3.html#K6" title="PRVECT_3:func.6">prod_NORM</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c18">v1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D6" target="_self" title="PRVECT_3:def.6">Def6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">Ix1</font> = <font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c20">Iy1</font> = <font color="Maroon" title="c17">y1</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="prvect_3.html#E19"><span class="lab"><font color="Green" title="E12">Lm4</font></span></a></span>;<br/>
<a NAME="E6:46_6"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">Ix1</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">X</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">Iy1</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="c2">Y</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>;<br/>
<span class="lab"><font color="Green" title="E29">A11</font></span>: <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> = 
<font color="Maroon" title="c14">I</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c17">y1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_6"><span class="lab"><font color="Green" title="E26">A8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c19">Ix1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c20">Iy1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E25">A7</font></span></a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">Iv</font> = <font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">Ix</font> = <font color="Maroon" title="c16">x1</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">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">Iy</font> = <font color="Maroon" title="c17">y1</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="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E30">A12</font></span>: <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="prvect_2.html#D12" title="PRVECT_2:def.12">PRVECT_2:def 12</a></span>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E12:46_6"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <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"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D11" title="PRVECT_2:def.11">PRVECT_2:def 11</a></span>;<br/>
<span class="lab"><font color="Green" title="E32">A14</font></span>: <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c18">v1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c16">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_6"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="topreal6.html#T11" title="TOPREAL6:th.11">TOPREAL6:11</a></span>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c16">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T77" title="RVSUM_1:th.77">RVSUM_1:77</a></span>
;<br/>
<a NAME="E14:46_6"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</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">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T86" title="RVSUM_1:th.86">RVSUM_1:86</a></span>;<br/>
<a NAME="E15:46_6"/>
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c16">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><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">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><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="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="prvect_2.html#D12" title="PRVECT_2:def.12">PRVECT_2:def 12</a></span>;<br/>
<a NAME="E16:46_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A16</font></span>: 
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c16">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</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">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span></span>)</span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:46_6"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="ref" href="square_1.html#D2" title="SQUARE_1:def.2">SQUARE_1:def 2</a></span>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> = 
<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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46_6"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
;<br/>
<a NAME="E18:46_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A17</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <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"><span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E19:46_6"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_6_5"/>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span><br/>

<a NAME="E2:46_6_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A18</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:46_6_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A19</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46_6"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">k1</font> = <font color="Maroon" title="c24">k</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"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_6_5"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E12:46_6"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:46_6_5"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
<span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="normsp_0.html#U1" title="NORMSP_0:sel.1">normF</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c25">k1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D11" title="PRVECT_2:def.11">PRVECT_2:def 11</a></span>;<br/>
<a NAME="E6:46_6_5"/><span class="lab"><font color="Green" title="E39">A21</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">Ix1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">X</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">Iy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">Y</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_6"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:46_6_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A22</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">Ix1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">Iy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_6_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:46_6_5_1"/>
( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">X</font> or  ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">X</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_6_5"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_6_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:46_6_5_1_1"/><span class="lab"><font color="Green" title="E41">A23</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span><br/><div class="add"><a NAME="E2:46_6_5_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">X</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D11" title="PRVECT_2:def.11">PRVECT_2:def 11</a></span>;<br/><a NAME="E3:46_6_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A24</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_6_5_1_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">k2</font> = <font color="Maroon" title="c25">k1</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">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_6_5_1_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a></span>;<br/><a NAME="E5:46_6_5_1_1"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
<font color="Maroon" title="c21">Iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Ix1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_6_5_1_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="txt" href="#E7:46_6_5"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="txt" href="#E7:46_6"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> = 
 the <a href="normsp_0.html#U1" title="NORMSP_0:sel.1">normF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c26">k2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_6_5"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_6_5_1_1"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="ref" href="prvect_2.html#D11" title="PRVECT_2:def.11">PRVECT_2:def 11</a></span>
<br/>.= 
<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">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_6_5_1_1"><span class="lab"><font color="Green" title="E42">A24</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
; <a class="txt" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_6_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:46_6_5_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">X</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:46_6_5_1_2"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
( <font color="Maroon" title="c26">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">X</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">n</font> )
 ;<br/><a NAME="E4:46_6_5_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">Y</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D11" title="PRVECT_2:def.11">PRVECT_2:def 11</a></span>;<br/><a NAME="E5:46_6_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A27</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_6_5_1_2"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c26">n</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="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_6_5_1_2"><span class="lab"><font color="Green" title="E41">A26</font></span></a></span>;<br/><a NAME="E7:46_6_5_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">Ix1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_6_5"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:46_6_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A28</font></span>: 
<font color="Maroon" title="c21">Iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Iy1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_6"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E3:46_6_5_1_2"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E7:46_6_5"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> = 
 the <a href="normsp_0.html#U1" title="NORMSP_0:sel.1">normF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <font color="Maroon" title="c27">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_6_5"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E3:46_6_5_1_2"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:46_6_5_1_2"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="prvect_2.html#D11" title="PRVECT_2:def.11">PRVECT_2:def 11</a></span>
<br/>.= 
<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">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_6_5_1_2"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="txt" href="#E3:46_6_5_1_2"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E12:46_6"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
; <a class="txt" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E20:46_6"/><span class="kw">then </span>
 <a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</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">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:46_6"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>;<br/>
<a NAME="E21:46_6"/><span class="kw">then </span>
 <a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c21">Iv</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</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">X</font>,<font color="Maroon" title="c22">Ix</font>)</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K12" title="PRVECT_2:func.12">normsequence</a> (<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c23">Iy</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="prvect_3.html#E26"><span class="lab"><font color="Green" title="E17">Lm6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:46_6"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c15">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46_6"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E11:46_6"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E16:46_6"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E4:46_6"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E2:46_6"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>; <a class="txt" 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">hence </span><a NAME="E19:46"/>
 for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
