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

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Z</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>  ex <font color="Olive" title="b2">I</font> being   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">Lipschitzian</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b2">I</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a>  &amp; (  for <font color="Olive" title="b3">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b3">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">u</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  for <font color="Olive" title="b4">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="Olive" title="b1">Y</font></span>)</span><br/>  for <font color="Olive" title="b5">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b5">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b4">y</font> ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">Y</font> be   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="lopban_1.html#V2" title="LOPBAN_1:attr.2">Lipschitzian</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</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; <font color="Olive" title="b1">I</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a>  &amp; (  for <font color="Olive" title="b2">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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">u</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span><br/>  for <font color="Olive" title="b4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <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">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b4">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b3">y</font> ) ) ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">CX</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">CY</font> =  <a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">CZ</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">J</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p3">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p4">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
( <font color="Maroon" title="c7">J</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span><br/>  for <font color="Olive" title="b2">y</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/>  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; <font color="Olive" title="b3">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 <font color="Maroon" title="c1">X</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_6.html#T6" onmouseover="rs('ndiff_6/T6')" onmouseout="rh()">NDIFF_6:6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">LXYZ</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">BXYZ</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">LYZ</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>;<br/>
<a NAME="E3:46"/>
 <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></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_1.html#K15" title="PRVECT_1:func.15">carr</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1"><a href="prvect_1.html#K12" title="PRVECT_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span><a href="prvect_1.html#K12" title="PRVECT_1:func.12">:]</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#K5" title="PRVECT_2:func.5">multop</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></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#K9" title="PRVECT_2:func.9">productnorm</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#T6" onmouseover="rs('prvect_2/T6')" onmouseout="rh()">PRVECT_2:6</a></span>;<br/>
<a NAME="E4:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 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#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T30" target="_self" onmouseover="rs('lopban14/T30')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E5:46"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
 <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <font color="Maroon" title="c3">Y</font></span>)</span>,<span class="p1"><a href="prvect_1.html#K12" title="PRVECT_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c3">Y</font></span>)</span></span><a href="prvect_1.html#K12" title="PRVECT_1:func.12">:]</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#K5" title="PRVECT_2:func.5">multop</a> <font color="Maroon" title="c3">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#K9" title="PRVECT_2:func.9">productnorm</a> <font color="Maroon" title="c3">Y</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#T6" onmouseover="rs('prvect_2/T6')" onmouseout="rh()">PRVECT_2:6</a></span>;<br/>
<a NAME="E6:46"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E7:46"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E8:46"/>
 <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></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_1.html#K15" title="PRVECT_1:func.15">carr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>,<span class="p1"><a href="prvect_1.html#K12" title="PRVECT_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span></span><a href="prvect_1.html#K12" title="PRVECT_1:func.12">:]</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#K5" title="PRVECT_2:func.5">multop</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></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#K9" title="PRVECT_2:func.9">productnorm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#T6" onmouseover="rs('prvect_2/T6')" onmouseout="rh()">PRVECT_2:6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E36">A6</font></span>:  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#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T29" target="_self" onmouseover="rs('lopban14/T29')" onmouseout="rh()">Th30</a></span>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T30" target="_self" onmouseover="rs('lopban14/T30')" onmouseout="rh()">Th31</a></span>
;<br/>
<a NAME="E10:46"/>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c11">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>) or <font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_2"/>
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:46_2"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
( <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c12">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c12">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E4:46_2"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c12">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E3:46_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:46_2"/>
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E11:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">I</font> = <font color="Maroon" title="c7">J</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T32" onmouseover="rs('funct_2/T32')" onmouseout="rh()">FUNCT_2:32</a></span>;<br/>
<a NAME="E13:46"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
 for <font color="Olive" title="b1">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> holds <br/> ( (  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b3">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b2">x</font> &amp; (  for <font color="Olive" title="b4">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="c3">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b4">y</font> ) ) ) &amp; <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> is   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI: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="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> &amp;  ex <font color="Olive" title="b2">If</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <br/>( <font color="Olive" title="b2">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">If</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c12">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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 <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b1">x</font> &amp; (  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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b3">y</font> ) ) ) &amp; <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> is   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R2" title="TARSKI: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="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> &amp;  ex <font color="Olive" title="b1">If</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">If</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> ) )</span><br/>

<a NAME="E1:46_3"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E2:46_3"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">f0</font> = <font color="Maroon" title="c12">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a>, <a class="ref" href="funct_2.html#T66" onmouseover="rs('funct_2/T66')" onmouseout="rh()">FUNCT_2:66</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">f1</font> = <font color="Maroon" title="c12">f</font> as   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">Lipschitzian</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" onmouseover="rs('lopban_1/D9')" onmouseout="rh()">LOPBAN_1:def 9</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:46_3"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b1">x</font> &amp; (  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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b3">y</font> ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> is   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R2" title="TARSKI: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="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> &amp;  ex <font color="Olive" title="b1">If</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">If</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c15">x</font> &amp; (  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="c3">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b2">y</font> ) )</span><br/>

<a NAME="E1:46_3_1"/>
<font color="Maroon" title="c14">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lopban10.html#K12" title="LOPBAN10:func.12">BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">g</font> = <font color="Maroon" title="c13">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> as   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#D11" onmouseover="rs('lopban10/D11')" onmouseout="rh()">LOPBAN10:def 11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">g</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c15">x</font> &amp; (  for <font color="Olive" title="b1">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="c3">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:46_3_1"/>
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c15">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="c3">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">y</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c17">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</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="c17">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c17">y</font></span><br/>

<span class="kw">thus </span><a NAME="E4:46_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</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="c17">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c15">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c17">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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_3"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<font color="Maroon" title="c7">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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"><span class="p3">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a>, <a class="ref" href="funct_2.html#T66" onmouseover="rs('funct_2/T66')" onmouseout="rh()">FUNCT_2:66</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">Jf</font> = <font color="Maroon" title="c7">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <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">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E2:46_3"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a>, <a class="ref" href="funct_2.html#T66" onmouseover="rs('funct_2/T66')" onmouseout="rh()">FUNCT_2:66</a></span>;<br/>
<a NAME="E8:46_3"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span><br/>  for <font color="Olive" title="b2">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> holds  <font color="Maroon" title="c15">Jf</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">z</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> holds  <font color="Maroon" title="c15">Jf</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Olive" title="b1">z</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c17">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">Jf</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c18">y</font>, <font color="Maroon" title="c19">px</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:46_3_2"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c19">px</font> &amp; <font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; <font color="Maroon" title="c19">px</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T2" onmouseover="rs('rlaffin3/T2')" onmouseout="rh()">RLAFFIN3:2</a>, <a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E36">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">y</font> = <font color="Maroon" title="c18">y</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">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:46_3_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:46_3_2"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<font color="Maroon" title="c19">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E2:46_3_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">cy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:46_3_2"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">cy</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c22">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds <br/><font color="Maroon" title="c22">cy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="card_3.html#D5" onmouseover="rs('card_3/D5')" onmouseout="rh()">CARD_3:def 5</a></span>;<br/>
<a NAME="E8:46_3_2"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c22">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_3_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">cy</font> = <font color="Maroon" title="c22">cy</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="#E7:46_3_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A18</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_3_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E8:46_3_2"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D11" onmouseover="rs('prvect_1/D11')" onmouseout="rh()">PRVECT_1:def 11</a></span>
;<br/>
<span class="lab"><font color="Green" title="E48">A19</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> = 
 <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"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
;<br/>
<a NAME="E12:46_3_2"/><span class="lab"><font color="Green" title="E49">A20</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_3_2"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_2_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:46_3_2_3"/>
( <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_2_3_1"><span class="kw">suppose </span></a><a NAME="E1:46_3_2_3_1"/>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">Jf</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font></span><br/><div class="add"><a NAME="E2:46_3_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI: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> <font color="Maroon" title="c3">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_3_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E12:46_3_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">i0</font> = <font color="Maroon" title="c16">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">Y</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c25">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E5:46_3_2_3_1"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
( <font color="Maroon" title="c25">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c21">x</font> &amp; (  for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/><a NAME="E6:46_3_2_3_1"/><span class="lab"><font color="Green" title="E51">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c24">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">L</font> = <font color="Maroon" title="c15">Jf</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c24">i0</font></span>)</span>,<font color="Maroon" title="c2">Z</font> ;<br/><a NAME="E8:46_3_2_3_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E6:46_3"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E9:46_3_2_3_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A23</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/><a NAME="E11:46_3_2_3_1"/><span class="lab"><font color="Green" title="E53">A24</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c20">y</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c24">i0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E12:46_3_2_3_1"/>
 for <font color="Olive" title="b1">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c20">y</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_2_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c20">y</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">w</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_3_2_3_1_2"/><span class="lab"><font color="Green" title="E54">A25</font></span>: 
<font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c20">y</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">w</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">w1</font> = <font color="Maroon" title="c27">w</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">w2</font> = <font color="Maroon" title="c27">w</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c24">i0</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_3_2_3_1_2"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E4:46_3_2_3_1_2"/><span class="lab"><font color="Green" title="E55">A26</font></span>: 
<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">w1</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="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c20">y</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">w2</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E5:46_3_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="lopban14.html#T31" target="_self" onmouseover="rs('lopban14/T31')" onmouseout="rh()">Th32</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c25">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c20">y</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">w2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3_2_3_1"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E4:46_3_2_3_1_2"><span class="lab"><font color="Green" title="E55">A26</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c20">y</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E13:46_3_2_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c20">y</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_3_2_3_1"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="#E10:46_3_2_3_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E11:46_3_2_3_1"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E14:46_3_2_3_1"/>
<font color="Maroon" title="c15">Jf</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E6:46_3_2_3_1"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="ref" href="lopban10.html#D6" onmouseover="rs('lopban10/D6')" onmouseout="rh()">LOPBAN10:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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_3_2_3_2"><span class="kw">suppose </span></a><a NAME="E1:46_3_2_3_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">Jf</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font></span><br/><div class="add"><a NAME="E2:46_3_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A28</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:46_3_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A29</font></span>: 
<font color="Maroon" title="c16">i</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="c20">y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46_3_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E10:46_3_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E4:46_3_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A30</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_3_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="finseq_1.html#T42" onmouseover="rs('finseq_1/T42')" onmouseout="rh()">FINSEQ_1:42</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">h</font> = <font color="Maroon" title="c15">Jf</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Z</font> ;<br/><span class="lab"><font color="Green" title="E53">A31</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_3_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E3:46_3_2_3_2"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="finseq_1.html#T42" onmouseover="rs('finseq_1/T42')" onmouseout="rh()">FINSEQ_1:42</a></span>
;<br/><a NAME="E7:46_3_2_3_2"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</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="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_2_3_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x1</font>, <font color="Maroon" title="c26">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c26">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x1</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="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x2</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c27">g1</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E2:46_3_2_3_2_2"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
( <font color="Maroon" title="c27">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c25">x1</font> &amp; (  for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c25">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">g2</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E4:46_3_2_3_2_2"/><span class="lab"><font color="Green" title="E55">A33</font></span>: 
( <font color="Maroon" title="c28">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c26">x2</font> &amp; (  for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c26">x2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">g2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c29">g12</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E6:46_3_2_3_2_2"/><span class="lab"><font color="Green" title="E56">A34</font></span>: 
( <font color="Maroon" title="c29">g12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c26">x2</font></span>)</span> &amp; (  for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c26">x2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">g12</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/>
<a NAME="E7:46_3_2_3_2_2"/><span class="lab"><font color="Green" title="E57">A35</font></span>: 
<font color="Maroon" title="c14">f1</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a> 
 
;<br/>
<a NAME="E8:46_3_2_3_2_2"/><span class="lab"><font color="Green" title="E58">A36</font></span>: 
<font color="Maroon" title="c29">g12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c25">x1</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="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c26">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E7:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E57">A35</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E59">A37</font></span>: <font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c26">x2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c26">x2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E6:46_3_2_3_2"><span class="lab"><font color="Green" title="E53">A31</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</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="c20">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c26">x2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E5:46_3_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E10:46_3_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E12:46_3_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E2:46_3_2_3_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c25">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c26">x2</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c20">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E8:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E58">A36</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c20">y</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="c28">g2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c20">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E4:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="lopban10.html#T48" onmouseover="rs('lopban10/T48')" onmouseout="rh()">LOPBAN10:48</a></span>
;<br/>
<span class="lab"><font color="Green" title="E60">A38</font></span>: <font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a>, <a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E6:46_3_2_3_2"><span class="lab"><font color="Green" title="E53">A31</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</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="c20">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c25">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E5:46_3_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E10:46_3_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E12:46_3_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E2:46_3_2_3_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c27">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c20">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a></span>
;<br/>
<font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E6:46_3_2_3_2"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</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="c20">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c26">x2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E5:46_3_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E10:46_3_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E12:46_3_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E2:46_3_2_3_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c28">g2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c20">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:46_3_2_3_2_2"/>
<font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c26">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x1</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="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="txt" href="#E10:46_3_2_3_2_2"><span class="lab"><font color="Green" title="E60">A38</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E8:46_3_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A40</font></span>: 
<font color="Maroon" title="c24">h</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a> 
 ;<br/><a NAME="E9:46_3_2_3_2"/>
 for <font color="Olive" title="b1">x0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x0</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_2_3_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c25">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x0</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c26">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c25">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x0</font></span>)</span></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c27">g1</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E2:46_3_2_3_2_3"/><span class="lab"><font color="Green" title="E55">A41</font></span>: 
( <font color="Maroon" title="c27">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c25">x0</font> &amp; (  for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c25">x0</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">g2</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E4:46_3_2_3_2_3"/><span class="lab"><font color="Green" title="E56">A42</font></span>: 
( <font color="Maroon" title="c28">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c25">x0</font></span>)</span> &amp; (  for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c25">x0</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">g2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/>
<a NAME="E5:46_3_2_3_2_3"/><span class="lab"><font color="Green" title="E57">A43</font></span>: 
<font color="Maroon" title="c28">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_3_2_3_2_3"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="ref" href="lopban_1.html#D5" onmouseover="rs('lopban_1/D5')" onmouseout="rh()">LOPBAN_1:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A44</font></span>: <font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c25">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c25">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E6:46_3_2_3_2"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</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="c20">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c25">x0</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E5:46_3_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E10:46_3_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E12:46_3_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E2:46_3_2_3_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c25">x0</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c20">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_3_2_3_2_3"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="txt" href="#E5:46_3_2_3_2_3"><span class="lab"><font color="Green" title="E57">A43</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c20">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2_3_2_3"><span class="lab"><font color="Green" title="E55">A41</font></span></a>, <a class="ref" href="lopban10.html#T49" onmouseover="rs('lopban10/T49')" onmouseout="rh()">LOPBAN10:49</a></span>
;<br/>
<font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E6:46_3_2_3_2"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</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="c20">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c25">x0</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E5:46_3_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E10:46_3_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E12:46_3_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E2:46_3_2_3_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c27">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c20">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_2_3_2_3"><span class="lab"><font color="Green" title="E55">A41</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:46_3_2_3_2_3"/>
<font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c25">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_3_2_3_2_3"><span class="lab"><font color="Green" title="E58">A44</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E10:46_3_2_3_2"/>
<font color="Maroon" title="c15">Jf</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c16">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_3_2_3_2"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E8:46_3_2_3_2"><span class="lab"><font color="Green" title="E54">A40</font></span></a>, <a class="ref" href="lopban_1.html#D5" onmouseover="rs('lopban_1/D5')" onmouseout="rh()">LOPBAN_1:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">If</font> = <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> as   <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="lopban10.html#D6" onmouseover="rs('lopban10/D6')" onmouseout="rh()">LOPBAN10:def 6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">K</font> = <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>;<br/>
<a NAME="E10:46_3"/><span class="lab"><font color="Green" title="E43">A46</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T33" onmouseover="rs('lopban_1/T33')" onmouseout="rh()">LOPBAN_1:33</a></span>;<br/>
<a NAME="E11:46_3"/><span class="lab"><font color="Green" title="E44">A47</font></span>: 
 for <font color="Olive" title="b1">z</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">If</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Olive" title="b1">z</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">If</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c18">z</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c18">z</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c19">py</font>, <font color="Maroon" title="c20">px</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:46_3_3"/><span class="lab"><font color="Green" title="E45">A48</font></span>: 
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c20">px</font> &amp; <font color="Maroon" title="c19">py</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; <font color="Maroon" title="c20">px</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="rlaffin3.html#T2" onmouseover="rs('rlaffin3/T2')" onmouseout="rh()">RLAFFIN3:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">py</font> = <font color="Maroon" title="c19">py</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">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:46_3_3"><span class="lab"><font color="Green" title="E45">A48</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:46_3_3"/><span class="lab"><font color="Green" title="E46">A49</font></span>: 
<font color="Maroon" title="c20">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c22">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E2:46_3_3"><span class="lab"><font color="Green" title="E45">A48</font></span></a>, <a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">cy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:46_3_3"/><span class="lab"><font color="Green" title="E47">A50</font></span>: 
( <font color="Maroon" title="c21">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">cy</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c23">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds <br/><font color="Maroon" title="c23">cy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_3"><span class="lab"><font color="Green" title="E45">A48</font></span></a>, <a class="ref" href="card_3.html#D5" onmouseover="rs('card_3/D5')" onmouseout="rh()">CARD_3:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">cy</font> = <font color="Maroon" title="c23">cy</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="#E7:46_3_3"><span class="lab"><font color="Green" title="E47">A50</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E10:46_3_3"/><span class="lab"><font color="Green" title="E48">A51</font></span>: 
( <font color="Maroon" title="c25">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c22">x1</font> &amp; (  for <font color="Olive" title="b1">p</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c22">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/>
<a NAME="E11:46_3_3"/><span class="lab"><font color="Green" title="E49">A52</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c21">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_3_3"><span class="lab"><font color="Green" title="E48">A51</font></span></a>, <a class="txt" href="#E2:46_3_3"><span class="lab"><font color="Green" title="E45">A48</font></span></a>, <a class="txt" href="#E5:46_3_3"><span class="lab"><font color="Green" title="E46">A49</font></span></a></span>;<br/>
<a NAME="E12:46_3_3"/><span class="lab"><font color="Green" title="E50">A53</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c21">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c22">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c21">py</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_3_3"><span class="lab"><font color="Green" title="E48">A51</font></span></a>, <a class="ref" href="lopban10.html#T45" onmouseover="rs('lopban10/T45')" onmouseout="rh()">LOPBAN10:45</a></span>;<br/>
<a NAME="E13:46_3_3"/><span class="lab"><font color="Green" title="E51">A54</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c21">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D7" onmouseover="rs('xxreal_0/D7')" onmouseout="rh()">XXREAL_0:def 7</a></span>;<br/>
<a NAME="E14:46_3_3"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c22">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T32" onmouseover="rs('lopban_1/T32')" onmouseout="rh()">LOPBAN_1:32</a></span>;<br/>
<a NAME="E15:46_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A55</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c22">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c21">py</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c22">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c21">py</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46_3_3"><span class="lab"><font color="Green" title="E51">A54</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E16:46_3_3"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c22">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c21">py</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c18">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_3_3"><span class="lab"><font color="Green" title="E45">A48</font></span></a>, <a class="txt" href="#E5:46_3_3"><span class="lab"><font color="Green" title="E46">A49</font></span></a>, <a class="ref" href="lopban14.html#T34" target="_self" onmouseover="rs('lopban14/T34')" onmouseout="rh()">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:46_3_3"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">If</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c18">z</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c18">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_3_3"><span class="lab"><font color="Green" title="E49">A52</font></span></a>, <a class="txt" href="#E12:46_3_3"><span class="lab"><font color="Green" title="E50">A53</font></span></a>, <a class="txt" href="#E15:46_3_3"><span class="lab"><font color="Green" title="E52">A55</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E12:46_3"/><span class="lab"><font color="Green" title="E45">A56</font></span>: 
<font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> is   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#D10" onmouseover="rs('lopban10/D10')" onmouseout="rh()">LOPBAN10:def 10</a>, <a class="ref" href="lopban_1.html#T33" onmouseover="rs('lopban_1/T33')" onmouseout="rh()">LOPBAN_1:33</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R2" title="TARSKI: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="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> &amp;  ex <font color="Olive" title="b1">If</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">If</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> ) )</span><br/>

<span class="kw">hence </span><a NAME="E13:46_3"/>
<font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R2" title="TARSKI: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="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#D11" onmouseover="rs('lopban10/D11')" onmouseout="rh()">LOPBAN10:def 11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">If</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">If</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">If1</font> = <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46_3"><span class="lab"><font color="Green" title="E45">A56</font></span></a>, <a class="ref" href="lopban10.html#D11" onmouseover="rs('lopban10/D11')" onmouseout="rh()">LOPBAN10:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">If2</font> = <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> as   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_3"><span class="lab"><font color="Green" title="E44">A47</font></span></a>, <a class="ref" href="lopban10.html#D10" onmouseover="rs('lopban10/D10')" onmouseout="rh()">LOPBAN10:def 10</a>, <a class="ref" href="lopban_1.html#T33" onmouseover="rs('lopban_1/T33')" onmouseout="rh()">LOPBAN_1:33</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c19">If2</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c19">If2</font> implies <font color="Maroon" title="c20">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:46_3_4"/>
<font color="Maroon" title="c20">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c19">If2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">z</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <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">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:46_3_4"/><span class="lab"><font color="Green" title="E46">A57</font></span>: 
( <font color="Maroon" title="c20">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">If2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c21">z</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="lopban10.html#K10" title="LOPBAN10:func.10">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )
 ;<br/><a NAME="E4:46_3_4"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c21">z</font> &amp;  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c21">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_3_4"><span class="lab"><font color="Green" title="E46">A57</font></span></a>, <a class="ref" href="lopban10.html#T35" onmouseover="rs('lopban10/T35')" onmouseout="rh()">LOPBAN10:35</a></span>;<br/><a NAME="E5:46_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A58</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c21">z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_3"><span class="lab"><font color="Green" title="E43">A46</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E6:46_3_4"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">If2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c21">z</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c21">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_3"><span class="lab"><font color="Green" title="E44">A47</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:46_3_4"/>
<font color="Maroon" title="c20">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_3_4"><span class="lab"><font color="Green" title="E46">A57</font></span></a>, <a class="txt" href="#E5:46_3_4"><span class="lab"><font color="Green" title="E47">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E17:46_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A59</font></span>: 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c19">If2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T45" onmouseover="rs('seq_4/T45')" onmouseout="rh()">SEQ_4:45</a></span>;<br/>
<a NAME="E18:46_3"/><span class="lab"><font color="Green" title="E47">A60</font></span>: 
<span class="p1">(<span class="default"><a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">If2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D1" onmouseover="rs('normsp_0/D1')" onmouseout="rh()">NORMSP_0:def 1</a></span>;<br/>
<a NAME="E19:46_3"/><span class="lab"><font color="Green" title="E48">A61</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:46_3"><span class="lab"><font color="Green" title="E46">A59</font></span></a>, <a class="txt" href="#E18:46_3"><span class="lab"><font color="Green" title="E47">A60</font></span></a>, <a class="ref" href="lopban10.html#T43" onmouseover="rs('lopban10/T43')" onmouseout="rh()">LOPBAN10:43</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">SLYZ</font> =  <a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">f1</font> = <font color="Maroon" title="c12">f</font> as   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">Lipschitzian</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" onmouseover="rs('lopban_1/D9')" onmouseout="rh()">LOPBAN_1:def 9</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c21">f1</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c21">f1</font> implies <font color="Maroon" title="c22">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:46_3_5"/>
<font color="Maroon" title="c22">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c21">f1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">x1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:46_3_5"/><span class="lab"><font color="Green" title="E49">A62</font></span>: 
( <font color="Maroon" title="c22">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c24">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E5:46_3_5"/><span class="lab"><font color="Green" title="E50">A63</font></span>: 
( <font color="Maroon" title="c24">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c23">x1</font> &amp; (  for <font color="Olive" title="b1">p</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/><a NAME="E6:46_3_5"/><span class="lab"><font color="Green" title="E51">A64</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T4" onmouseover="rs('normsp_1/T4')" onmouseout="rh()">NORMSP_1:4</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_5_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c24">g</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c24">g</font> implies <font color="Maroon" title="c25">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:46_3_5_1"/>
<font color="Maroon" title="c25">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c24">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">py</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</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">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:46_3_5_1"/><span class="lab"><font color="Green" title="E52">A65</font></span>: 
( <font color="Maroon" title="c25">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c26">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">Y</font> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">py</font> <a href="lopban10.html#K10" title="LOPBAN10:func.10">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">pz</font> = <font color="Maroon" title="c26">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/><a NAME="E5:46_3_5_1"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c26">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">If2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c27">pz</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3_5"><span class="lab"><font color="Green" title="E50">A63</font></span></a></span>;<br/><a NAME="E6:46_3_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A66</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c26">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c27">pz</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#T45" onmouseover="rs('lopban10/T45')" onmouseout="rh()">LOPBAN10:45</a></span>;<br/><a NAME="E7:46_3_5_1"/><span class="lab"><font color="Green" title="E54">A67</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c26">py</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c27">pz</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T34" target="_self" onmouseover="rs('lopban14/T34')" onmouseout="rh()">Th35</a></span>;<br/><a NAME="E8:46_3_5_1"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c26">py</font> &amp;  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c26">py</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_3_5_1"><span class="lab"><font color="Green" title="E52">A65</font></span></a>, <a class="ref" href="lopban10.html#T35" onmouseover="rs('lopban10/T35')" onmouseout="rh()">LOPBAN10:35</a></span>;<br/><a NAME="E9:46_3_5_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c26">py</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_3_5"><span class="lab"><font color="Green" title="E51">A64</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E10:46_3_5_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c27">pz</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_3_5"><span class="lab"><font color="Green" title="E51">A64</font></span></a>, <a class="txt" href="#E7:46_3_5_1"><span class="lab"><font color="Green" title="E54">A67</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E11:46_3_5_1"/>
<font color="Maroon" title="c25">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_3_5_1"><span class="lab"><font color="Green" title="E52">A65</font></span></a>, <a class="txt" href="#E6:46_3_5_1"><span class="lab"><font color="Green" title="E53">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E8:46_3_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A68</font></span>: 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c24">g</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T45" onmouseover="rs('seq_4/T45')" onmouseout="rh()">SEQ_4:45</a></span>;<br/><a NAME="E9:46_3_5"/><span class="lab"><font color="Green" title="E53">A69</font></span>: 
<span class="p1">(<span class="default"><a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c23">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_3_5"><span class="lab"><font color="Green" title="E50">A63</font></span></a>, <a class="ref" href="normsp_0.html#D1" onmouseover="rs('normsp_0/D1')" onmouseout="rh()">NORMSP_0:def 1</a></span>;<br/><a NAME="E10:46_3_5"/><span class="lab"><font color="Green" title="E54">A70</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c23">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#T43" onmouseover="rs('lopban10/T43')" onmouseout="rh()">LOPBAN10:43</a>, <a class="txt" href="#E8:46_3_5"><span class="lab"><font color="Green" title="E52">A68</font></span></a>, <a class="txt" href="#E9:46_3_5"><span class="lab"><font color="Green" title="E53">A69</font></span></a></span>;<br/><a NAME="E11:46_3_5"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c23">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_3_5"><span class="lab"><font color="Green" title="E49">A62</font></span></a>, <a class="txt" href="#E6:46_3_5"><span class="lab"><font color="Green" title="E51">A64</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E12:46_3_5"/>
<font color="Maroon" title="c22">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_3_5"><span class="lab"><font color="Green" title="E49">A62</font></span></a>, <a class="txt" href="#E10:46_3_5"><span class="lab"><font color="Green" title="E54">A70</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E22:46_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A71</font></span>: 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c21">f1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">If1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T45" onmouseover="rs('seq_4/T45')" onmouseout="rh()">SEQ_4:45</a></span>;<br/>
<a NAME="E23:46_3"/><span class="lab"><font color="Green" title="E50">A72</font></span>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D1" onmouseover="rs('normsp_0/D1')" onmouseout="rh()">NORMSP_0:def 1</a></span>;<br/>
<a NAME="E24:46_3"/>
<span class="p1">(<span class="default"><a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c21">f1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T30" onmouseover="rs('lopban_1/T30')" onmouseout="rh()">LOPBAN_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:46_3"/>
 ex <font color="Olive" title="b1">If</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c12">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">If</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:46_3"><span class="lab"><font color="Green" title="E48">A61</font></span></a>, <a class="txt" href="#E22:46_3"><span class="lab"><font color="Green" title="E49">A71</font></span></a>, <a class="txt" href="#E23:46_3"><span class="lab"><font color="Green" title="E50">A72</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E14:46"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T114" onmouseover="rs('funct_2/T114')" onmouseout="rh()">FUNCT_2:114</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">I</font> = <font color="Maroon" title="c11">I</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" onmouseover="rs('funct_2/T6')" onmouseout="rh()">FUNCT_2:6</a></span>;<br/>
<a NAME="E16:46"/><span class="lab"><font color="Green" title="E39">A73</font></span>: 
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> holds  <font color="Maroon" title="c12">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">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f1</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="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f2</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c13">f1</font>, <font color="Maroon" title="c14">f2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</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="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f2</font></span>)</span></span><br/>

<a NAME="E1:46_4"/>
 for <font color="Olive" title="b1">z</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">f2</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f2</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">z</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">z</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#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">f2</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f2</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c16">py</font>, <font color="Maroon" title="c17">px</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:46_4_1"/><span class="lab"><font color="Green" title="E40">A74</font></span>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c17">px</font> &amp; <font color="Maroon" title="c16">py</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; <font color="Maroon" title="c17">px</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="rlaffin3.html#T2" onmouseover="rs('rlaffin3/T2')" onmouseout="rh()">RLAFFIN3:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">py</font> = <font color="Maroon" title="c16">py</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">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:46_4_1"><span class="lab"><font color="Green" title="E40">A74</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:46_4_1"/><span class="lab"><font color="Green" title="E41">A75</font></span>: 
<font color="Maroon" title="c17">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E2:46_4_1"><span class="lab"><font color="Green" title="E40">A74</font></span></a>, <a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">g1</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E7:46_4_1"/><span class="lab"><font color="Green" title="E42">A76</font></span>: 
( <font color="Maroon" title="c20">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f1</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">x1</font> &amp; (  for <font color="Olive" title="b1">p</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">g2</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E9:46_4_1"/><span class="lab"><font color="Green" title="E43">A77</font></span>: 
( <font color="Maroon" title="c21">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f2</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">x1</font> &amp; (  for <font color="Olive" title="b1">ppy</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">ppy</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">ppy</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">g12</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E11:46_4_1"/><span class="lab"><font color="Green" title="E44">A78</font></span>: 
( <font color="Maroon" title="c22">g12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">f2</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">x1</font> &amp; (  for <font color="Olive" title="b1">p</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">f2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g12</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<a NAME="E12:46_4_1"/><span class="lab"><font color="Green" title="E45">A79</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f2</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c18">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_4_1"><span class="lab"><font color="Green" title="E40">A74</font></span></a>, <a class="txt" href="#E5:46_4_1"><span class="lab"><font color="Green" title="E41">A75</font></span></a>, <a class="txt" href="#E9:46_4_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a></span>;<br/>
<a NAME="E13:46_4_1"/><span class="lab"><font color="Green" title="E46">A80</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">f2</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g12</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c18">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_4_1"><span class="lab"><font color="Green" title="E40">A74</font></span></a>, <a class="txt" href="#E5:46_4_1"><span class="lab"><font color="Green" title="E41">A75</font></span></a>, <a class="txt" href="#E11:46_4_1"><span class="lab"><font color="Green" title="E44">A78</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">py</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">f2</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">py</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">f2</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">py</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#T48" onmouseover="rs('lopban10/T48')" onmouseout="rh()">LOPBAN10:48</a></span>
<br/>.= 
<font color="Maroon" title="c22">g12</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c18">py</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:46_4_1"><span class="lab"><font color="Green" title="E44">A78</font></span></a>, <a class="ref" href="lopban_1.html#T35" onmouseover="rs('lopban_1/T35')" onmouseout="rh()">LOPBAN_1:35</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:46_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">f2</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f2</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_4_1"><span class="lab"><font color="Green" title="E40">A74</font></span></a>, <a class="txt" href="#E5:46_4_1"><span class="lab"><font color="Green" title="E41">A75</font></span></a>, <a class="txt" href="#E7:46_4_1"><span class="lab"><font color="Green" title="E42">A76</font></span></a>, <a class="txt" href="#E9:46_4_1"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="txt" href="#E12:46_4_1"><span class="lab"><font color="Green" title="E45">A79</font></span></a>, <a class="txt" href="#E13:46_4_1"><span class="lab"><font color="Green" title="E46">A80</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:46_4"/>
<font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</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="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#T48" onmouseover="rs('lopban10/T48')" onmouseout="rh()">LOPBAN10:48</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E17:46"/>
 for <font color="Olive" title="b1">f1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span><br/>  for <font color="Olive" title="b2">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c12">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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f1</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c13">f1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c12">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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c14">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span></span><br/>



<a NAME="E1:46_5"/>
 for <font color="Olive" title="b1">z</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">z</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">z</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#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c16">py</font>, <font color="Maroon" title="c17">px</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:46_5_1"/><span class="lab"><font color="Green" title="E40">A82</font></span>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c17">px</font> &amp; <font color="Maroon" title="c16">py</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; <font color="Maroon" title="c17">px</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="rlaffin3.html#T2" onmouseover="rs('rlaffin3/T2')" onmouseout="rh()">RLAFFIN3:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">py</font> = <font color="Maroon" title="c16">py</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">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:46_5_1"><span class="lab"><font color="Green" title="E40">A82</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">ppx</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:46_5_1"/><span class="lab"><font color="Green" title="E41">A83</font></span>: 
<font color="Maroon" title="c17">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">ppx</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E2:46_5_1"><span class="lab"><font color="Green" title="E40">A82</font></span></a>, <a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">g1</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E7:46_5_1"/><span class="lab"><font color="Green" title="E42">A84</font></span>: 
( <font color="Maroon" title="c20">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f1</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">ppx</font> &amp; (  for <font color="Olive" title="b1">p</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">ppx</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">g2</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E9:46_5_1"/><span class="lab"><font color="Green" title="E43">A85</font></span>: 
( <font color="Maroon" title="c21">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">ppx</font> &amp; (  for <font color="Olive" title="b1">p</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">ppx</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<a NAME="E10:46_5_1"/><span class="lab"><font color="Green" title="E44">A86</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c18">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_5_1"><span class="lab"><font color="Green" title="E40">A82</font></span></a>, <a class="txt" href="#E5:46_5_1"><span class="lab"><font color="Green" title="E41">A83</font></span></a>, <a class="txt" href="#E9:46_5_1"><span class="lab"><font color="Green" title="E43">A85</font></span></a></span>;<br/>
<font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">ppx</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">py</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c19">ppx</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c18">py</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#T49" onmouseover="rs('lopban10/T49')" onmouseout="rh()">LOPBAN10:49</a></span>
<br/>.= 
<font color="Maroon" title="c21">g2</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c18">py</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46_5_1"><span class="lab"><font color="Green" title="E43">A85</font></span></a>, <a class="ref" href="lopban_1.html#T36" onmouseover="rs('lopban_1/T36')" onmouseout="rh()">LOPBAN_1:36</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:46_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_5_1"><span class="lab"><font color="Green" title="E40">A82</font></span></a>, <a class="txt" href="#E5:46_5_1"><span class="lab"><font color="Green" title="E41">A83</font></span></a>, <a class="txt" href="#E7:46_5_1"><span class="lab"><font color="Green" title="E42">A84</font></span></a>, <a class="txt" href="#E10:46_5_1"><span class="lab"><font color="Green" title="E44">A86</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:46_5"/>
<font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#T49" onmouseover="rs('lopban10/T49')" onmouseout="rh()">LOPBAN10:49</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">I</font> = <font color="Maroon" title="c12">I</font> as   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:46"><span class="lab"><font color="Green" title="E39">A73</font></span></a>, <a class="ref" href="lopban_1.html#D5" onmouseover="rs('lopban_1/D5')" onmouseout="rh()">LOPBAN_1:def 5</a>, <a class="ref" href="vectsp_1.html#D20" onmouseover="rs('vectsp_1/D20')" onmouseout="rh()">VECTSP_1:def 20</a></span>;<br/>
<a NAME="E19:46"/><span class="lab"><font color="Green" title="E40">A88</font></span>: 
 for <font color="Olive" title="b1">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  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="c3">Y</font></span>)</span><br/>  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">y</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c14">u</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c14">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  for <font color="Olive" title="b1">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="c3">Y</font></span>)</span><br/>  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<a NAME="E1:46_6"/>
 ex <font color="Olive" title="b1">Iu</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <br/>( <font color="Olive" title="b1">Iu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c14">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">Iu</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:46_6"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c14">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="c3">Y</font></span>)</span><br/>  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">y</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c15">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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 <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</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="c15">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c16">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</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="c15">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">y</font></span><br/>



<a NAME="E3:46_6"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c16">x</font> &amp; (  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="c3">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b2">y</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:46_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</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="c15">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c15">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E20:46"/><span class="lab"><font color="Green" title="E41">A89</font></span>: 
 for <font color="Olive" title="b1">If</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">If</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="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> holds <br/> ex <font color="Olive" title="b2">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">f</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="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> &amp; <font color="Olive" title="b1">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">f</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">If</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">If</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="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> implies  ex <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">f</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="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c14">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_7"/><span class="lab"><font color="Green" title="E42">A90</font></span>: 
<font color="Maroon" title="c14">If</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="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">f</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="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c14">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Iu</font> = <font color="Maroon" title="c14">If</font> as   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#D11" onmouseover="rs('lopban10/D11')" onmouseout="rh()">LOPBAN10:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">Lu</font> = <font color="Maroon" title="c14">If</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_7"><span class="lab"><font color="Green" title="E42">A90</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; (  for <font color="Olive" title="b3">py</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">Y</font></span>)</span> holds  <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b3">py</font> ) );<br/>
<span class="kw">set </span><font color="Maroon" title="c17">LYZ</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>;<br/>
<a NAME="E4:46_7"/><span class="lab"><font color="Green" title="E43">A91</font></span>: 
 for <font color="Olive" title="b1">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">x1</font>,<font color="Olive" title="b1">w</font>]</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">py</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">Y</font></span>)</span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">py</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> );<br/>
<a NAME="E1:46_7_1"/><span class="lab"><font color="Green" title="E44">A92</font></span>: 
 for <font color="Olive" title="b1">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>  ex <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">v</font>,<font color="Olive" title="b1">w</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c20">If</font> = <font color="Maroon" title="c19">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E2:46_7_1_1"/>
<font color="Maroon" title="c15">Iu</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c20">If</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Z</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:46_7_1_1"/>
 ex <font color="Olive" title="b1">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">v</font>,<font color="Olive" title="b1">w</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">consider </span><font color="Maroon" title="c19">g1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E3:46_7_1"/><span class="lab"><font color="Green" title="E45">A93</font></span>: 
 for <font color="Olive" title="b1">py</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">py</font>,<font color="Maroon" title="c19">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">py</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:46_7_1"><span class="lab"><font color="Green" title="E44">A92</font></span></a>);<br/></span>
<a NAME="E4:46_7_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">Y</font><br/>  for <font color="Olive" title="b2">py</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds  <font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">py</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">Y</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">py</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds  <font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Olive" title="b1">py</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c20">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c21">py</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c20">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c22">cy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:46_7_1_2"/><span class="lab"><font color="Green" title="E46">A94</font></span>: 
( <font color="Maroon" title="c21">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">cy</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c22">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds <br/><font color="Maroon" title="c22">cy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="card_3.html#D5" onmouseover="rs('card_3/D5')" onmouseout="rh()">CARD_3:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">cy</font> = <font color="Maroon" title="c22">cy</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="#E2:46_7_1_2"><span class="lab"><font color="Green" title="E46">A94</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A95</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> = 
 <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"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
;<br/>
<a NAME="E5:46_7_1_2"/>
<font color="Maroon" title="c20">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">Y</font>
 
;<br/>
<a NAME="E6:46_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</font> <a href="tarski.html#R2" title="TARSKI: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> <font color="Maroon" title="c3">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:46_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A96</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E8:46_7_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E9:46_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E7:46_7_1_2"><span class="lab"><font color="Green" title="E48">A96</font></span></a></span>;<br/>
<a NAME="E10:46_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_7_1_2"><span class="lab"><font color="Green" title="E48">A96</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">i0</font> = <font color="Maroon" title="c20">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_7_1_2"><span class="lab"><font color="Green" title="E47">A95</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">z</font> = <font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E13:46_7_1_2"/><span class="lab"><font color="Green" title="E49">A97</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c24">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c20">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">L</font> = <font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c20">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font> ;<br/>
<span class="lab"><font color="Green" title="E50">A98</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c24">i0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>
;<br/>
<a NAME="E16:46_7_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">Iu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E17:46_7_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">Iu</font>
 
;<br/>
<a NAME="E18:46_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A99</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:46_7_1_2"><span class="lab"><font color="Green" title="E50">A98</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E19:46_7_1_2"/>
 for <font color="Olive" title="b1">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">t</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_7_1_2_3"/><span class="lab"><font color="Green" title="E52">A100</font></span>: 
<font color="Maroon" title="c27">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">t</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">t1</font> = <font color="Maroon" title="c27">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c24">i0</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">t2</font> = <font color="Maroon" title="c27">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_7_1_2_3"><span class="lab"><font color="Green" title="E52">A100</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E4:46_7_1_2_3"/><span class="lab"><font color="Green" title="E53">A101</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">t2</font>,<font color="Maroon" title="c19">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">t2</font></span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_7_1"><span class="lab"><font color="Green" title="E45">A93</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">t</font> = 
<font color="Maroon" title="c15">Iu</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">t1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c19">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">t2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_7_1_2_3"><span class="lab"><font color="Green" title="E53">A101</font></span></a>, <a class="ref" href="lopban14.html#T31" target="_self" onmouseover="rs('lopban14/T31')" onmouseout="rh()">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E20:46_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c24">i0</font>,<font color="Maroon" title="c25">z</font>)</span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46_7_1_2"><span class="lab"><font color="Green" title="E49">A97</font></span></a>, <a class="txt" href="#E18:46_7_1_2"><span class="lab"><font color="Green" title="E51">A99</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:46_7_1_2"/>
<font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">py</font>)</span>)</span> is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c20">i</font></span>)</span>,<font color="Maroon" title="c2">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46_7_1_2"><span class="lab"><font color="Green" title="E49">A97</font></span></a>, <a class="ref" href="lopban10.html#D6" onmouseover="rs('lopban10/D6')" onmouseout="rh()">LOPBAN10:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">g1</font> = <font color="Maroon" title="c19">g1</font> as   <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#D6" onmouseover="rs('lopban10/D6')" onmouseout="rh()">LOPBAN10:def 6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">K</font> = <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">Lu</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>;<br/>
<a NAME="E6:46_7_1"/><span class="lab"><font color="Green" title="E46">A102</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">Lu</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T4" onmouseover="rs('normsp_1/T4')" onmouseout="rh()">NORMSP_1:4</a></span>;<br/>
<a NAME="E7:46_7_1"/>
 for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">Lu</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Olive" title="b1">py</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">py</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c22">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">Lu</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c22">py</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c23">cy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:46_7_1_3"/><span class="lab"><font color="Green" title="E47">A103</font></span>: 
( <font color="Maroon" title="c22">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">cy</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c23">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds <br/><font color="Maroon" title="c23">cy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="card_3.html#D5" onmouseover="rs('card_3/D5')" onmouseout="rh()">CARD_3:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">cy</font> = <font color="Maroon" title="c23">cy</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="#E2:46_7_1_3"><span class="lab"><font color="Green" title="E47">A103</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">z</font> = <font color="Maroon" title="c22">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E5:46_7_1_3"/><span class="lab"><font color="Green" title="E48">A104</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c22">py</font>,<font color="Maroon" title="c20">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c22">py</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_7_1"><span class="lab"><font color="Green" title="E45">A93</font></span></a></span>;<br/>
<a NAME="E6:46_7_1_3"/><span class="lab"><font color="Green" title="E49">A105</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c25">z</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">Lu</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c25">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#T45" onmouseover="rs('lopban10/T45')" onmouseout="rh()">LOPBAN10:45</a></span>;<br/>
<a NAME="E7:46_7_1_3"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c22">py</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c25">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T34" target="_self" onmouseover="rs('lopban14/T34')" onmouseout="rh()">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:46_7_1_3"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c22">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">Lu</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c22">py</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_7_1_3"><span class="lab"><font color="Green" title="E48">A104</font></span></a>, <a class="txt" href="#E6:46_7_1_3"><span class="lab"><font color="Green" title="E49">A105</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">g1</font> = <font color="Maroon" title="c20">g1</font> as   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_7_1"><span class="lab"><font color="Green" title="E46">A102</font></span></a>, <a class="ref" href="lopban10.html#D10" onmouseover="rs('lopban10/D10')" onmouseout="rh()">LOPBAN10:def 10</a>, <a class="ref" href="xreal_1.html#T127" onmouseover="rs('xreal_1/T127')" onmouseout="rh()">XREAL_1:127</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">g2</font> = <font color="Maroon" title="c22">g1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#D11" onmouseover="rs('lopban10/D11')" onmouseout="rh()">LOPBAN10:def 11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c23">g2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c23">g2</font>]</span><br/>

<a NAME="E10:46_7_1"/>
 for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">py</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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">py</font></span><br/>

<a NAME="E1:46_7_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c24">py</font>,<font color="Maroon" title="c22">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">py</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_7_1"><span class="lab"><font color="Green" title="E45">A93</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:46_7_1_4"/>
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c18">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">py</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E11:46_7_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c23">g2</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">consider </span><font color="Maroon" title="c18">f0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E6:46_7"/><span class="lab"><font color="Green" title="E44">A106</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E4:46_7"><span class="lab"><font color="Green" title="E43">A91</font></span></a>);<br/></span>
<a NAME="E7:46_7"/><span class="lab"><font color="Green" title="E45">A107</font></span>: 
 for <font color="Olive" title="b1">x0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">y0</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">Y</font></span>)</span> holds  <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y0</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">x0</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x0</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">y0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y0</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">Y</font></span>)</span> holds  <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y0</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x0</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">y0</font></span><br/>

<a NAME="E1:46_7_2"/>
 ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x0</font> &amp; <font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; (  for <font color="Olive" title="b3">py</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">Y</font></span>)</span> holds  <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b3">py</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_7"><span class="lab"><font color="Green" title="E44">A106</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:46_7_2"/>
 for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x0</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x0</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">py</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E8:46_7"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</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="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c20">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</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="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x2</font></span>)</span></span><br/>

<a NAME="E1:46_7_3"/>
 for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">py</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x2</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">py</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">py</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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x2</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c22">cy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:46_7_3_1"/><span class="lab"><font color="Green" title="E46">A108</font></span>: 
( <font color="Maroon" title="c21">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">cy</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c22">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds <br/><font color="Maroon" title="c22">cy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="card_3.html#D5" onmouseover="rs('card_3/D5')" onmouseout="rh()">CARD_3:def 5</a></span>;<br/>
<a NAME="E3:46_7_3_1"/><span class="lab"><font color="Green" title="E47">A109</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c22">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_7_3_1"><span class="lab"><font color="Green" title="E46">A108</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">cy</font> = <font color="Maroon" title="c22">cy</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="#E2:46_7_3_1"><span class="lab"><font color="Green" title="E46">A108</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E48">A110</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">py</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_7_3_1"><span class="lab"><font color="Green" title="E46">A108</font></span></a>, <a class="txt" href="#E3:46_7_3_1"><span class="lab"><font color="Green" title="E47">A109</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D11" onmouseover="rs('prvect_1/D11')" onmouseout="rh()">PRVECT_1:def 11</a></span>
;<br/>
<a NAME="E6:46_7_3_1"/><span class="lab"><font color="Green" title="E49">A111</font></span>: 
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_7"><span class="lab"><font color="Green" title="E45">A107</font></span></a></span>;<br/>
<a NAME="E7:46_7_3_1"/><span class="lab"><font color="Green" title="E50">A112</font></span>: 
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c20">x2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x2</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_7"><span class="lab"><font color="Green" title="E45">A107</font></span></a></span>;<br/>
<a NAME="E8:46_7_3_1"/><span class="lab"><font color="Green" title="E51">A113</font></span>: 
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_7"><span class="lab"><font color="Green" title="E45">A107</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E52">A114</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">z1</font> = <font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">z2</font> = <font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c20">x2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">z12</font> = <font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E13:46_7_3_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" onmouseover="rs('finseq_1/T4')" onmouseout="rh()">FINSEQ_1:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">j</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">py</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_7_3_1"><span class="lab"><font color="Green" title="E48">A110</font></span></a>, <a class="txt" href="#E9:46_7_3_1"><span class="lab"><font color="Green" title="E52">A114</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E15:46_7_3_1"/><span class="lab"><font color="Green" title="E53">A115</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c27">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_7_3_1"><span class="lab"><font color="Green" title="E48">A110</font></span></a>, <a class="ref" href="finseq_1.html#T42" onmouseover="rs('finseq_1/T42')" onmouseout="rh()">FINSEQ_1:42</a></span>;<br/>
<span class="lab"><font color="Green" title="E54">A116</font></span>: <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c27">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c27">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:46_7_3_1"><span class="lab"><font color="Green" title="E53">A115</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<span class="lab"><font color="Green" title="E55">A117</font></span>: <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c20">x2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c27">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c27">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:46_7_3_1"><span class="lab"><font color="Green" title="E53">A115</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<span class="lab"><font color="Green" title="E56">A118</font></span>: <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c27">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c27">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:46_7_3_1"><span class="lab"><font color="Green" title="E53">A115</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">L</font> = <font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c27">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span> as   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:46_7_3_1"><span class="lab"><font color="Green" title="E53">A115</font></span></a>, <a class="ref" href="lopban10.html#D6" onmouseover="rs('lopban10/D6')" onmouseout="rh()">LOPBAN10:def 6</a></span>;<br/>
<a NAME="E20:46_7_3_1"/>
<font color="Maroon" title="c28">L</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E21:46_7_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x2</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:46_7_3_1"><span class="lab"><font color="Green" title="E54">A116</font></span></a>, <a class="txt" href="#E17:46_7_3_1"><span class="lab"><font color="Green" title="E55">A117</font></span></a>, <a class="txt" href="#E18:46_7_3_1"><span class="lab"><font color="Green" title="E56">A118</font></span></a>, <a class="txt" href="#E6:46_7_3_1"><span class="lab"><font color="Green" title="E49">A111</font></span></a>, <a class="txt" href="#E7:46_7_3_1"><span class="lab"><font color="Green" title="E50">A112</font></span></a>, <a class="txt" href="#E8:46_7_3_1"><span class="lab"><font color="Green" title="E51">A113</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:46_7_3"/>
<font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</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="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#T48" onmouseover="rs('lopban10/T48')" onmouseout="rh()">LOPBAN10:48</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E9:46_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A119</font></span>: 
<font color="Maroon" title="c18">f0</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a> 
 
;<br/>
<a NAME="E10:46_7"/>
 for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c20">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span></span><br/>



<a NAME="E1:46_7_4"/>
 for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b1">py</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">py</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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c22">cy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:46_7_4_1"/><span class="lab"><font color="Green" title="E47">A120</font></span>: 
( <font color="Maroon" title="c21">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">cy</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c22">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds <br/><font color="Maroon" title="c22">cy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="card_3.html#D5" onmouseover="rs('card_3/D5')" onmouseout="rh()">CARD_3:def 5</a></span>;<br/>
<a NAME="E3:46_7_4_1"/><span class="lab"><font color="Green" title="E48">A121</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c22">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_7_4_1"><span class="lab"><font color="Green" title="E47">A120</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">cy</font> = <font color="Maroon" title="c22">cy</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="#E2:46_7_4_1"><span class="lab"><font color="Green" title="E47">A120</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E49">A122</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">py</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_7_4_1"><span class="lab"><font color="Green" title="E47">A120</font></span></a>, <a class="txt" href="#E3:46_7_4_1"><span class="lab"><font color="Green" title="E48">A121</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D11" onmouseover="rs('prvect_1/D11')" onmouseout="rh()">PRVECT_1:def 11</a></span>
;<br/>
<a NAME="E6:46_7_4_1"/><span class="lab"><font color="Green" title="E50">A123</font></span>: 
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_7"><span class="lab"><font color="Green" title="E45">A107</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E51">A124</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">z1</font> = <font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">z2</font> = <font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E10:46_7_4_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" onmouseover="rs('finseq_1/T4')" onmouseout="rh()">FINSEQ_1:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">j</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">py</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 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="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_7_4_1"><span class="lab"><font color="Green" title="E49">A122</font></span></a>, <a class="txt" href="#E7:46_7_4_1"><span class="lab"><font color="Green" title="E51">A124</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:46_7_4_1"/><span class="lab"><font color="Green" title="E52">A125</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="prvect_2.html#K7" title="PRVECT_2:func.7">.</a> <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_7_4_1"><span class="lab"><font color="Green" title="E49">A122</font></span></a>, <a class="ref" href="finseq_1.html#T42" onmouseover="rs('finseq_1/T42')" onmouseout="rh()">FINSEQ_1:42</a></span>;<br/>
<span class="lab"><font color="Green" title="E53">A126</font></span>: <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c26">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c26">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46_7_4_1"><span class="lab"><font color="Green" title="E52">A125</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<span class="lab"><font color="Green" title="E54">A127</font></span>: <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c26">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T32" target="_self" onmouseover="rs('lopban14/T32')" onmouseout="rh()">Th33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c26">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46_7_4_1"><span class="lab"><font color="Green" title="E52">A125</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">L</font> = <font color="Maroon" title="c15">Iu</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<font color="Maroon" title="c26">j</font>,<font color="Maroon" title="c24">z1</font>)</span>)</span> as   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:46_7_4_1"><span class="lab"><font color="Green" title="E52">A125</font></span></a>, <a class="ref" href="lopban10.html#D6" onmouseover="rs('lopban10/D6')" onmouseout="rh()">LOPBAN10:def 6</a></span>;<br/>
<a NAME="E16:46_7_4_1"/>
<font color="Maroon" title="c27">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D5" onmouseover="rs('lopban_1/D5')" onmouseout="rh()">LOPBAN_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:46_7_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Maroon" title="c21">py</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_7"><span class="lab"><font color="Green" title="E45">A107</font></span></a>, <a class="txt" href="#E6:46_7_4_1"><span class="lab"><font color="Green" title="E50">A123</font></span></a>, <a class="txt" href="#E13:46_7_4_1"><span class="lab"><font color="Green" title="E53">A126</font></span></a>, <a class="txt" href="#E14:46_7_4_1"><span class="lab"><font color="Green" title="E54">A127</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:46_7_4"/>
<font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#T49" onmouseover="rs('lopban10/T49')" onmouseout="rh()">LOPBAN10:49</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">f0</font> = <font color="Maroon" title="c18">f0</font> as   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46_7"><span class="lab"><font color="Green" title="E46">A119</font></span></a>, <a class="ref" href="lopban_1.html#D5" onmouseover="rs('lopban_1/D5')" onmouseout="rh()">LOPBAN_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">K</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E13:46_7"/><span class="lab"><font color="Green" title="E47">A128</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> &amp; (  for <font color="Olive" title="b1">z1</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">z1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Olive" title="b1">z1</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#D10" onmouseover="rs('lopban10/D10')" onmouseout="rh()">LOPBAN10:def 10</a></span>;<br/>
<a NAME="E14:46_7"/>
 for <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c22">fx1</font> = <font color="Maroon" title="c19">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x1</font> as   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#D11" onmouseover="rs('lopban10/D11')" onmouseout="rh()">LOPBAN10:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">fx2</font> = <font color="Maroon" title="c19">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> ;<br/>
<a NAME="E3:46_7_5"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T4" onmouseover="rs('normsp_1/T4')" onmouseout="rh()">NORMSP_1:4</a></span>;<br/>
<a NAME="E4:46_7_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A129</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T127" onmouseover="rs('xreal_1/T127')" onmouseout="rh()">XREAL_1:127</a>, <a class="txt" href="#E13:46_7"><span class="lab"><font color="Green" title="E47">A128</font></span></a></span>;<br/>
<a NAME="E5:46_7_5"/><span class="lab"><font color="Green" title="E49">A130</font></span>: 
 for <font color="Olive" title="b1">px</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">Y</font></span>)</span> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">fx1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">px</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Olive" title="b1">px</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">py</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#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c3">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">fx1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c24">py</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c25">cy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:46_7_5_1"/><span class="lab"><font color="Green" title="E50">A131</font></span>: 
( <font color="Maroon" title="c24">py</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">cy</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c25">cy</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</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_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> holds <br/><font color="Maroon" title="c25">cy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="card_3.html#D5" onmouseover="rs('card_3/D5')" onmouseout="rh()">CARD_3:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">cy</font> = <font color="Maroon" title="c25">cy</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="#E2:46_7_5_1"><span class="lab"><font color="Green" title="E50">A131</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">z1</font> = <font color="Maroon" title="c24">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T33" target="_self" onmouseover="rs('lopban14/T33')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E5:46_7_5_1"/><span class="lab"><font color="Green" title="E51">A132</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Iu</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c27">z1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c27">z1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46_7"><span class="lab"><font color="Green" title="E47">A128</font></span></a></span>;<br/>
<a NAME="E6:46_7_5_1"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c24">py</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c27">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T34" target="_self" onmouseover="rs('lopban14/T34')" onmouseout="rh()">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:46_7_5_1"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">fx1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c24">py</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_7"><span class="lab"><font color="Green" title="E45">A107</font></span></a>, <a class="txt" href="#E5:46_7_5_1"><span class="lab"><font color="Green" title="E51">A132</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_5_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c22">fx1</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c22">fx1</font> implies <font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:46_7_5_2"/>
<font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c22">fx1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">py</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</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">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:46_7_5_2"/><span class="lab"><font color="Green" title="E50">A133</font></span>: 
( <font color="Maroon" title="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">fx1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c25">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">Y</font> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">py</font> <a href="lopban10.html#K10" title="LOPBAN10:func.10">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )
 ;<br/><a NAME="E4:46_7_5_2"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c25">py</font> &amp;  <a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c25">py</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_7_5_2"><span class="lab"><font color="Green" title="E50">A133</font></span></a>, <a class="ref" href="lopban10.html#T35" onmouseover="rs('lopban10/T35')" onmouseout="rh()">LOPBAN10:35</a></span>;<br/><a NAME="E5:46_7_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A134</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c25">py</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_7_5"><span class="lab"><font color="Green" title="E48">A129</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E6:46_7_5_2"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">fx1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c25">py</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban10.html#K11" title="LOPBAN10:func.11">NrProduct</a> <font color="Maroon" title="c25">py</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_7_5"><span class="lab"><font color="Green" title="E49">A130</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:46_7_5_2"/>
<font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_7_5_2"><span class="lab"><font color="Green" title="E50">A133</font></span></a>, <a class="txt" href="#E5:46_7_5_2"><span class="lab"><font color="Green" title="E51">A134</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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_7_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A135</font></span>: 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c22">fx1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T45" onmouseover="rs('seq_4/T45')" onmouseout="rh()">SEQ_4:45</a></span>;<br/>
<a NAME="E8:46_7_5"/>
<span class="p1">(<span class="default"><a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">fx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D1" onmouseover="rs('normsp_0/D1')" onmouseout="rh()">NORMSP_0:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:46_7_5"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">x1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">K</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_7_5"><span class="lab"><font color="Green" title="E50">A135</font></span></a>, <a class="ref" href="lopban10.html#T43" onmouseover="rs('lopban10/T43')" onmouseout="rh()">LOPBAN10:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">f0</font> = <font color="Maroon" title="c19">f0</font> as   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">Lipschitzian</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46_7"><span class="lab"><font color="Green" title="E47">A128</font></span></a>, <a class="ref" href="lopban_1.html#D8" onmouseover="rs('lopban_1/D8')" onmouseout="rh()">LOPBAN_1:def 8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c21">f0</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">f0</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="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c14">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">f0</font> )</span><br/>

<span class="kw">thus </span><a NAME="E16:46_7"/>
<font color="Maroon" title="c21">f0</font> <a href="tarski.html#R2" title="TARSKI: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="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" onmouseover="rs('lopban_1/D9')" onmouseout="rh()">LOPBAN_1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">f0</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c22">f</font> = <font color="Maroon" title="c21">f0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" onmouseover="rs('lopban_1/D9')" onmouseout="rh()">LOPBAN_1:def 9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">L</font> = <font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">f</font> as   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban10.html#D11" onmouseover="rs('lopban10/D11')" onmouseout="rh()">LOPBAN10:def 11</a></span>;<br/>
<a NAME="E19:46_7"/>
 for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span> holds  <font color="Maroon" title="c23">L</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Iu</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">z</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#K10" title="PRVECT_2:func.10">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">L</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Iu</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">z</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c25">py</font>, <font color="Maroon" title="c26">px</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:46_7_6"/><span class="lab"><font color="Green" title="E48">A136</font></span>: 
( <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">px</font> &amp; <font color="Maroon" title="c25">py</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Maroon" title="c3">Y</font></span>)</span> &amp; <font color="Maroon" title="c26">px</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="rlaffin3.html#T2" onmouseover="rs('rlaffin3/T2')" onmouseout="rh()">RLAFFIN3:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">py</font> = <font color="Maroon" title="c25">py</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">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:46_7_6"><span class="lab"><font color="Green" title="E48">A136</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:46_7_6"/><span class="lab"><font color="Green" title="E49">A137</font></span>: 
<font color="Maroon" title="c26">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c28">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E2:46_7_6"><span class="lab"><font color="Green" title="E48">A136</font></span></a>, <a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c29">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c30">g1</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E7:46_7_6"/><span class="lab"><font color="Green" title="E50">A138</font></span>: 
( <font color="Maroon" title="c28">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">x2</font> &amp; <font color="Maroon" title="c21">f0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">g1</font> &amp; (  for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <font color="Maroon" title="c15">Iu</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c29">x2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_7"><span class="lab"><font color="Green" title="E44">A106</font></span></a></span>;<br/>
<a NAME="E8:46_7_6"/><span class="lab"><font color="Green" title="E51">A139</font></span>: 
<font color="Maroon" title="c15">Iu</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">g1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c27">py</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_7_6"><span class="lab"><font color="Green" title="E48">A136</font></span></a>, <a class="txt" href="#E5:46_7_6"><span class="lab"><font color="Green" title="E49">A137</font></span></a>, <a class="txt" href="#E7:46_7_6"><span class="lab"><font color="Green" title="E50">A138</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c31">g</font> being   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font><span class="kw"> such that </span><br/><a NAME="E10:46_7_6"/><span class="lab"><font color="Green" title="E52">A140</font></span>: 
( <font color="Maroon" title="c31">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c28">x1</font> &amp; (  for <font color="Olive" title="b1">py</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">Y</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">py</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c28">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">g</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Olive" title="b1">py</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:46"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:46_7_6"/>
<font color="Maroon" title="c23">L</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Iu</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c24">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_7_6"><span class="lab"><font color="Green" title="E48">A136</font></span></a>, <a class="txt" href="#E5:46_7_6"><span class="lab"><font color="Green" title="E49">A137</font></span></a>, <a class="txt" href="#E7:46_7_6"><span class="lab"><font color="Green" title="E50">A138</font></span></a>, <a class="txt" href="#E8:46_7_6"><span class="lab"><font color="Green" title="E51">A139</font></span></a>, <a class="txt" href="#E10:46_7_6"><span class="lab"><font color="Green" title="E52">A140</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E20:46_7"/><span class="kw">then </span>
<font color="Maroon" title="c23">L</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">Iu</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E21:46_7"/>
<font color="Maroon" title="c14">If</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">f0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E21:46"/>
 for <font color="Olive" title="b1">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:46"><span class="lab"><font color="Green" title="E40">A88</font></span></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">I</font> as   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">Lipschitzian</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Z</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D8" onmouseover="rs('lopban_1/D8')" onmouseout="rh()">LOPBAN_1:def 8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">I</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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; <font color="Maroon" title="c14">I</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a>  &amp; (  for <font color="Olive" title="b1">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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">u</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  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="c3">Y</font></span>)</span><br/>  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <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">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">y</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E23:46"/>
( <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="#E2:46"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E20:46"><span class="lab"><font color="Green" title="E41">A89</font></span></a>, <a class="ref" href="funct_1.html#T52" onmouseover="rs('funct_1/T52')" onmouseout="rh()">FUNCT_1:52</a>, <a class="ref" href="funct_2.html#T10" onmouseover="rs('funct_2/T10')" onmouseout="rh()">FUNCT_2:10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a>  &amp; (  for <font color="Olive" title="b1">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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">u</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  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="c3">Y</font></span>)</span><br/>  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <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">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">y</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E24:46"/>
( <font color="Maroon" title="c14">I</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a>  &amp; (  for <font color="Olive" title="b1">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="lopban10.html#K18" title="LOPBAN10:func.18">R_NormSpace_of_BoundedMultilinearOperators</a> (<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">Z</font>)</span>)</span>)</span>)</span> holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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">u</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  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="c3">Y</font></span>)</span><br/>  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <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">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="lopban10.html#K19" title="LOPBAN10:func.19">.</a> <font color="Olive" title="b2">y</font> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:46"><span class="lab"><font color="Green" title="E40">A88</font></span></a>, <a class="ref" href="ndiff_7.html#T7" onmouseover="rs('ndiff_7/T7')" onmouseout="rh()">NDIFF_7:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
